%------------------------------------------------------------------------------
% File     : ITP285^3 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00107_006275
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0094_VEBT_SuccPredImperative_00107_006275 [Des22]

% Status   : ContradictoryAxioms
% Rating   : 0.88 v9.0.0, 0.90 v8.2.0, 0.85 v8.1.0
% Syntax   : Number of formulae    : 12547 (4797 unt;2269 typ;   0 def)
%            Number of atoms       : 31545 (13423 equ;   4 cnn)
%            Maximal formula atoms :   71 (   3 avg)
%            Number of connectives : 129008 (3401   ~; 485   |;2406   &;109262   @)
%                                         (   0 <=>;13454  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   7 avg)
%            Number of types       :  239 ( 238 usr)
%            Number of type conns  : 8455 (8455   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 2036 (2031 usr;  99 con; 0-4 aty)
%            Number of variables   : 30455 (2795   ^;26633   !;1027   ?;30455   :)
% 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 19:47:57.435
%------------------------------------------------------------------------------
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    array_of_list_int: list_int > heap_T1346037964561226099ay_int ).

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

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

thf(sy_c_Array__Time_Oupd_001_Eo,type,
    array_upd_o: nat > $o > array_o > heap_T5660665574680485309rray_o ).

thf(sy_c_Array__Time_Oupd_001t__Int__Oint,type,
    array_upd_int: nat > int > array_int > heap_T1346037964561226099ay_int ).

thf(sy_c_Array__Time_Oupd_001t__Nat__Onat,type,
    array_upd_nat: nat > nat > array_nat > heap_T3836121109492952855ay_nat ).

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

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

thf(sy_c_Assertions_Oprecise_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    precis1518117471652397933_VEBTi: ( list_VEBT_VEBTi > array_VEBT_VEBTi > assn ) > $o ).

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

thf(sy_c_Assertions_Osnga__assn_001_Eo,type,
    snga_assn_o: array_o > list_o > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Int__Oint,type,
    snga_assn_int: array_int > list_int > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: array_nat > list_nat > assn ).

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

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

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_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

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

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

thf(sy_c_Bit__Operations_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__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__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_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

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__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_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_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > int > int ).

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__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > int > int ).

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

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

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

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

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

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

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

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > 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_OSup__class_OSup_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    comple6471460710946744121eger_o: set_Co5460287733524608044eger_o > code_integer > code_integer > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    comple7687260386943045147_int_o: set_int_int_o > int > int > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Int__Oint_M_Eo_J,type,
    comple6496622788309502864_int_o: set_int_o > int > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    comple3592611370556534995_nat_o: set_nat_nat_o > nat > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    comple4350791933526045961_num_o: set_nat_num_o > nat > num > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
    comple8317665133742190828_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Real__Oreal_M_Eo_J,type,
    comple3015195443809154064real_o: set_real_o > real > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    comple3806919086088850358_nat_o: set_set_nat_o > set_nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    comple5511761119775097859_nat_o: set_VEBT_VEBT_nat_o > vEBT_VEBT > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    comple3152016690827447324VEBT_o: set_VEBT_VEBT_o > vEBT_VEBT > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
    comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).

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__Option__Ooption_I_Eo_J,type,
    comple4490649148004324043tion_o: set_option_o > option_o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple8455683388168444585et_nat: set_option_set_nat > option_set_nat ).

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

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
    comple90263536869209701_set_o: set_set_o > set_o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    comple739944243200306918nteger: set_set_Code_integer > set_Code_integer ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    comple8424636186594484919omplex: set_set_complex > set_complex ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Int__Oint_J,type,
    comple3221217463730067765et_int: set_set_int > set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    comple8404747032580312297st_nat: set_set_list_nat > set_list_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    comple2203973573673791208nteger: set_se4344029326803248219nteger > set_Pr4811707699266497531nteger ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    comple5382143125604098054nt_int: set_se6260736226359567993nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    comple1400524285282149784at_num: set_se4826145725398303499at_num > set_Pr6200539531224447659at_num ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    comple9061401370350521660BT_nat: set_se3932177096832370463BT_nat > set_Pr7556676689462069481BT_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
    comple3096694443085538997t_real: set_set_real > set_real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    comple2820511241208326657T_VEBT: set_set_VEBT_VEBT > set_VEBT_VEBT ).

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

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

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

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

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

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__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_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

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__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

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

thf(sy_c_Finite__Set_OFpow_001_Eo,type,
    finite_Fpow_o: set_o > set_set_o ).

thf(sy_c_Finite__Set_OFpow_001t__Code____Numeral__Ointeger,type,
    finite1532502677820914807nteger: set_Code_integer > set_set_Code_integer ).

thf(sy_c_Finite__Set_OFpow_001t__Complex__Ocomplex,type,
    finite_Fpow_complex: set_complex > set_set_complex ).

thf(sy_c_Finite__Set_OFpow_001t__Int__Oint,type,
    finite_Fpow_int: set_int > set_set_int ).

thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
    finite_Fpow_nat: set_nat > set_set_nat ).

thf(sy_c_Finite__Set_OFpow_001t__Real__Oreal,type,
    finite_Fpow_real: set_real > set_set_real ).

thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).

thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
    finite_card_list_nat: set_list_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Numeral____Type__Onum0,type,
    finite6454714172617411596l_num0: set_Numeral_num0 > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
    finite_card_literal: set_literal > nat ).

thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Code____Numeral__Ointeger,type,
    finite6017078050557962740nteger: set_Code_integer > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
    finite4001608067531595151d_enat: set_Extended_enat > $o ).

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

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

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Assertions__Oassn,type,
    minus_minus_assn: assn > assn > assn ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
    minus_minus_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Assertions__Oassn,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Assertions__Oassn,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,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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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__Rat__Orat_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__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Real__Oreal_001t__Int__Oint,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Real__Oreal_001t__Rat__Orat,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,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,
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thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,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__Heap__Oarray_It__Int__Oint_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Heap__Oarray_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Int__Oint,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    hoare_6480275734082232733on_nat: assn > heap_T5317711798761887292on_nat > ( list_option_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,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__Extended____Nat__Oenat,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

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

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

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

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

thf(sy_c_If_001t__Option__Ooption_I_Eo_J,type,
    if_option_o: $o > option_o > option_o > option_o ).

thf(sy_c_If_001t__Option__Ooption_It__Int__Oint_J,type,
    if_option_int: $o > option_int > option_int > option_int ).

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

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

thf(sy_c_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_opt6109864365331422477at_nat: $o > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Real__Oreal_J,type,
    if_option_real: $o > option_real > option_real > option_real ).

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__Option__Ooption_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    if_option_VEBT_VEBTi: $o > option_VEBT_VEBTi > option_VEBT_VEBTi > option_VEBT_VEBTi ).

thf(sy_c_If_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
    if_option_VEBT_VEBT: $o > option_VEBT_VEBT > option_VEBT_VEBT > option_VEBT_VEBT ).

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__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_I_Eo_J,type,
    if_set_o: $o > set_o > set_o > set_o ).

thf(sy_c_If_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    if_set_Code_integer: $o > set_Code_integer > set_Code_integer > set_Code_integer ).

thf(sy_c_If_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    if_set_complex: $o > set_complex > set_complex > set_complex ).

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__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__Set__Oset_It__Real__Oreal_J,type,
    if_set_real: $o > set_real > set_real > set_real ).

thf(sy_c_If_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    if_set_VEBT_VEBT: $o > set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

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

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_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_Osemilattice__neutr__order_001t__Assertions__Oassn,type,
    semila8603258263270017530r_assn: ( assn > assn > assn ) > assn > ( assn > assn > $o ) > ( assn > assn > $o ) > $o ).

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_Osemilattice__neutr__order_001t__Set__Oset_I_Eo_J,type,
    semila2554085542299052326_set_o: ( set_o > set_o > set_o ) > set_o > ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Int__Oint_J,type,
    semila6712789903965657268et_int: ( set_int > set_int > set_int ) > set_int > ( set_int > set_int > $o ) > ( set_int > set_int > $o ) > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
    semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Real__Oreal_J,type,
    semila4459961161675019956t_real: ( set_real > set_real > set_real ) > set_real > ( set_real > set_real > $o ) > ( set_real > set_real > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    sup_su2386495712988733984eger_o: ( code_integer > code_integer > $o ) > ( code_integer > code_integer > $o ) > code_integer > code_integer > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    sup_sup_complex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    sup_sup_int_int_o: ( int > int > $o ) > ( int > int > $o ) > int > int > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Int__Oint_M_Eo_J,type,
    sup_sup_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    sup_sup_nat_num_o: ( nat > num > $o ) > ( nat > num > $o ) > nat > num > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
    sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Real__Oreal_M_Eo_J,type,
    sup_sup_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    sup_su2199749269212604124_nat_o: ( vEBT_VEBT > nat > $o ) > ( vEBT_VEBT > nat > $o ) > vEBT_VEBT > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    sup_sup_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
    sup_sup_o: $o > $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__Int__Oint,type,
    sup_sup_int: int > int > int ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Assertions__Oassn_J,type,
    sup_sup_option_assn: option_assn > option_assn > option_assn ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Nat__Onat_J,type,
    sup_sup_option_nat: option_nat > option_nat > option_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_su3598758113090618626et_nat: option_set_nat > option_set_nat > option_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
    sup_sup_rat: rat > rat > rat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
    sup_sup_real: real > real > real ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    sup_su848401254843788991nteger: set_Code_integer > set_Code_integer > set_Code_integer ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    sup_sup_set_complex: set_complex > set_complex > set_complex ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    sup_su4489774667511045786d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
    sup_sup_set_int: set_int > set_int > set_int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_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_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
    sup_sup_set_num: set_num > set_num > set_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    sup_su3575575879904067535nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    sup_su2042722026077122175at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    sup_su6061789376821058069BT_nat: set_Pr7556676689462069481BT_nat > set_Pr7556676689462069481BT_nat > set_Pr7556676689462069481BT_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
    sup_sup_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
    sup_sup_set_real: set_real > set_real > set_real ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    sup_su6272177626956685416T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Int__Oint,type,
    lattices_Min_int: ( int > int > $o ) > set_int > int ).

thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Nat__Onat,type,
    lattices_Min_nat: ( nat > nat > $o ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Num__Onum,type,
    lattices_Min_num: ( num > num > $o ) > set_num > num ).

thf(sy_c_Lattices__Big_Olinorder_OMin_001t__Rat__Orat,type,
    lattices_Min_rat: ( rat > rat > $o ) > set_rat > rat ).

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__Extended____Nat__Oenat,type,
    lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).

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_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__Num__Onum,type,
    lattic2632268341708553362er_num: ( code_integer > num ) > set_Code_integer > code_integer ).

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__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
    lattic8440615504127631091l_real: ( real > real ) > 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_List_Oappend_001_Eo,type,
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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_Oappend_001t__Num__Onum,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Oappend_001t__Rat__Orat,type,
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thf(sy_c_List_Oappend_001t__Real__Oreal,type,
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thf(sy_c_List_Oappend_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Oappend_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Obind_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_List_Obind_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Obind_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Obind_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Obind_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Obind_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Obind_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oconcat_001_Eo,type,
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thf(sy_c_List_Oconcat_001t__Int__Oint,type,
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thf(sy_c_List_Oconcat_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Oconcat_001t__Real__Oreal,type,
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thf(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Odistinct_001_Eo,type,
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thf(sy_c_List_Odistinct_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odistinct_001t__Num__Onum,type,
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thf(sy_c_List_Odistinct_001t__Rat__Orat,type,
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thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
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thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Odrop_001_Eo,type,
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thf(sy_c_List_Odrop_001t__Int__Oint,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Num__Onum,type,
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thf(sy_c_List_Odrop_001t__Rat__Orat,type,
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thf(sy_c_List_Odrop_001t__Real__Oreal,type,
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thf(sy_c_List_Odrop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Odrop_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Odrop_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oenumerate_001_Eo,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__Num__Onum,type,
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thf(sy_c_List_Oenumerate_001t__Real__Oreal,type,
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thf(sy_c_List_Oenumerate_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldl_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_List_Ofoldl_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ofoldl_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Ofoldl_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldl_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldl_001t__List__Olist_It__Nat__Onat_J_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ofoldl_001t__List__Olist_It__Real__Oreal_J_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ofoldl_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_List_Ofoldl_001t__Rat__Orat_001t__Rat__Orat,type,
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thf(sy_c_List_Ofoldl_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Ofoldl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_List_Ofoldr_001_Eo_001t__Int__Oint,type,
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thf(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Olast_001_Eo,type,
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thf(sy_c_List_Olast_001t__Int__Oint,type,
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thf(sy_c_List_Olast_001t__Nat__Onat,type,
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thf(sy_c_List_Olast_001t__Real__Oreal,type,
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thf(sy_c_List_Olast_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olast_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001_Eo_001_Eo,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Rat__Orat,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
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thf(sy_c_List_Olinorder__class_Oinsort__key_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001_Eo,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Int__Oint,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Num__Onum,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Rat__Orat,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_OCons_001_Eo,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Num__Onum,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Olist_OCons_001t__Rat__Orat,type,
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thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_ONil_001_Eo,type,
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thf(sy_c_List_Olist_ONil_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_ONil_001t__Num__Onum,type,
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thf(sy_c_List_Olist_ONil_001t__Rat__Orat,type,
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thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_ONil_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_ONil_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__Code____Numeral__Ointeger_001t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Rat__Orat,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_I_Eo_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Int__Oint_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Real__Oreal_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__Num__Onum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    map_re7303296592222247787t_real: ( real > produc7716430852924023517t_real ) > list_real > list_P3644420460460130531t_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
    map_re731699941522841299real_o: ( real > product_prod_real_o ) > list_real > list_P3595434254542482545real_o ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    map_re3533924840734919879l_real: ( real > produc2422161461964618553l_real ) > list_real > list_P8689742595348180415l_real ).

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

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

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

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_real_VEBT_VEBTi: ( real > vEBT_VEBTi ) > list_real > list_VEBT_VEBTi ).

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

thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
    map_set_nat_real: ( set_nat > real ) > list_set_nat > list_real ).

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

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

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

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

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

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Nat__Onat_J,type,
    map_VE3614494202555519440st_nat: ( vEBT_VEBT > list_nat ) > list_VEBT_VEBT > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    map_VE92017342910548129VEBT_o: ( vEBT_VEBT > list_P3126845725202233233VEBT_o ) > list_VEBT_VEBT > list_l9063449193509804439VEBT_o ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    map_VE4456067000678548787BT_nat: ( vEBT_VEBT > list_P7037539587688870467BT_nat ) > list_VEBT_VEBT > list_l6108278773223375443BT_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    map_VE6455195641817765263T_real: ( vEBT_VEBT > list_P2623026923184700063T_real ) > list_VEBT_VEBT > list_l1703466484006042159T_real ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    map_VE6710161768887062461T_VEBT: ( vEBT_VEBT > list_P7413028617227757229T_VEBT ) > list_VEBT_VEBT > list_l9025237890452396851T_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__List__Olist_It__Real__Oreal_J,type,
    map_VE755361874264402988t_real: ( vEBT_VEBT > list_real ) > list_VEBT_VEBT > list_list_real ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    map_VE5619196432070217417T_VEBT: ( vEBT_VEBT > produc8025551001238799321T_VEBT ) > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    map_VE1210512188453230445T_VEBT: ( vEBT_VEBT > produc3757001726724277373T_VEBT ) > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    map_VE6666121349297532579BT_nat: ( vEBT_VEBT > produc9072475918466114483BT_nat ) > list_VEBT_VEBT > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).

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

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

thf(sy_c_List_Olist_Oset_001_Eo,type,
    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,
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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_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_I_Eo_Mt__Real__Oreal_J,type,
    set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).

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__Nat__Onat_M_Eo_J,type,
    set_Pr1291962091234853352_nat_o: list_P7333126701944960589_nat_o > set_Pr3149072824959771635_nat_o ).

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

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
    set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).

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__Nat__Onat_J,type,
    set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).

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__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_I_Eo_J,type,
    set_set_o2: list_set_o > set_set_o ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Int__Oint_J,type,
    set_set_int2: list_set_int > set_set_int ).

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__Set__Oset_It__Real__Oreal_J,type,
    set_set_real2: list_set_real > set_set_real ).

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

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

thf(sy_c_List_Olist_Osize__list_001_Eo,type,
    size_list_o: ( $o > nat ) > list_o > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Int__Oint,type,
    size_list_int: ( int > nat ) > list_int > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
    size_list_nat: ( nat > nat ) > list_nat > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Real__Oreal,type,
    size_list_real: ( real > nat ) > list_real > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Set__Oset_It__Nat__Onat_J,type,
    size_list_set_nat: ( set_nat > nat ) > list_set_nat > nat ).

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

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

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

thf(sy_c_List_Olist__update_001t__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_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__Real__Oreal,type,
    list_update_real: list_real > nat > real > list_real ).

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

thf(sy_c_List_Olist__update_001t__VEBT____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_On__lists_001_Eo,type,
    n_lists_o: nat > list_o > list_list_o ).

thf(sy_c_List_On__lists_001t__Int__Oint,type,
    n_lists_int: nat > list_int > list_list_int ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_On__lists_001t__Real__Oreal,type,
    n_lists_real: nat > list_real > list_list_real ).

thf(sy_c_List_On__lists_001t__VEBT____Definitions__OVEBT,type,
    n_lists_VEBT_VEBT: nat > list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > nat > code_integer ).

thf(sy_c_List_Onth_001t__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__Option__Ooption_It__Nat__Onat_J,type,
    nth_option_nat: list_option_nat > nat > option_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    nth_Pr2304437835452373666nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger ).

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,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    nth_Pr7767817059697521252t_real: list_P3644420460460130531t_real > nat > produc7716430852924023517t_real ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__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__Nat__Onat_J,type,
    nth_Pr6911489093701683181Ti_nat: list_P659468882601404396Ti_nat > nat > produc3881548065746020326Ti_nat ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr8725177398587324397T_VEBT: list_P5988454224134618948T_VEBT > nat > produc2810682830582626868T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: list_P2623026923184700063T_real > nat > produc5170161368751668367T_real ).

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

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

thf(sy_c_List_Onth_001t__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__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi ).

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

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

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

thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Oproduct_001_Eo_001t__Real__Oreal,type,
    product_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc8792966785426426881nteger: list_Code_integer > list_Code_integer > list_P5578671422887162913nteger ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
    product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    product_nat_real: list_nat > list_real > list_P3644420460460130531t_real ).

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

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

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

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

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct__lists_001_Eo,type,
    product_lists_o: list_list_o > list_list_o ).

thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
    product_lists_int: list_list_int > list_list_int ).

thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Oproduct__lists_001t__Real__Oreal,type,
    product_lists_real: list_list_real > list_list_real ).

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,
    remove1_int: int > list_int > list_int ).

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__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    remove8878500798450800835od_o_o: product_prod_o_o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o ).

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__Set__Oset_It__Nat__Onat_J,type,
    remove1_set_nat: set_nat > list_set_nat > list_set_nat ).

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_001_Eo,type,
    replicate_o: nat > $o > list_o ).

thf(sy_c_List_Oreplicate_001t__Code____Numeral__Ointeger,type,
    replic7707675349574490269nteger: nat > code_integer > list_Code_integer ).

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

thf(sy_c_List_Oreplicate_001t__List__Olist_I_Eo_J,type,
    replicate_list_o: nat > list_o > list_list_o ).

thf(sy_c_List_Oreplicate_001t__List__Olist_It__Int__Oint_J,type,
    replicate_list_int: nat > list_int > list_list_int ).

thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
    replicate_list_nat: nat > list_nat > list_list_nat ).

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

thf(sy_c_List_Oreplicate_001t__Num__Onum,type,
    replicate_num: nat > num > list_num ).

thf(sy_c_List_Oreplicate_001t__Rat__Orat,type,
    replicate_rat: nat > rat > list_rat ).

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

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

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

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

thf(sy_c_List_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__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    sorted4470838345478584340od_o_o: ( product_prod_o_o > product_prod_o_o > $o ) > list_P4002435161011370285od_o_o > $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__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__Code____Numeral__Ointeger,type,
    take_Code_integer: nat > list_Code_integer > list_Code_integer ).

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__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__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

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__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    zip_Co3543743374963494515nteger: list_Code_integer > list_Code_integer > list_P5578671422887162913nteger ).

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__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__Nat__Onat_001t__Num__Onum,type,
    zip_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Real__Oreal,type,
    zip_nat_real: list_nat > list_real > list_P3644420460460130531t_real ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

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__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__Nat__Onat,type,
    zip_VEBT_VEBTi_nat: list_VEBT_VEBTi > list_nat > list_P659468882601404396Ti_nat ).

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_Map_Ograph_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    graph_5282091004195177018nteger: ( code_integer > option_Code_integer ) > set_Pr4811707699266497531nteger ).

thf(sy_c_Map_Ograph_001t__Int__Oint_001t__Int__Oint,type,
    graph_int_int: ( int > option_int ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__Nat__Onat,type,
    graph_nat_nat: ( nat > option_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__Num__Onum,type,
    graph_nat_num: ( nat > option_num ) > set_Pr6200539531224447659at_num ).

thf(sy_c_Map_Ograph_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    graph_VEBT_VEBT_nat: ( vEBT_VEBT > option_nat ) > set_Pr7556676689462069481BT_nat ).

thf(sy_c_Map_Omap__upds_001_Eo_001_Eo,type,
    map_upds_o_o: ( $o > option_o ) > list_o > list_o > $o > option_o ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__Int__Oint,type,
    map_upds_o_int: ( $o > option_int ) > list_o > list_int > $o > option_int ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__Nat__Onat,type,
    map_upds_o_nat: ( $o > option_nat ) > list_o > list_nat > $o > option_nat ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__Num__Onum,type,
    map_upds_o_num: ( $o > option_num ) > list_o > list_num > $o > option_num ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__Real__Oreal,type,
    map_upds_o_real: ( $o > option_real ) > list_o > list_real > $o > option_real ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up3026452068722249431_VEBTi: ( $o > option_VEBT_VEBTi ) > list_o > list_VEBT_VEBTi > $o > option_VEBT_VEBTi ).

thf(sy_c_Map_Omap__upds_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    map_upds_o_VEBT_VEBT: ( $o > option_VEBT_VEBT ) > list_o > list_VEBT_VEBT > $o > option_VEBT_VEBT ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Int__Oint,type,
    map_upds_int_int: ( int > option_int ) > list_int > list_int > int > option_int ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Nat__Onat,type,
    map_upds_int_nat: ( int > option_nat ) > list_int > list_nat > int > option_nat ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Num__Onum,type,
    map_upds_int_num: ( int > option_num ) > list_int > list_num > int > option_num ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_up4628693354496902621at_nat: ( int > option4927543243414619207at_nat ) > list_int > list_P6011104703257516679at_nat > int > option4927543243414619207at_nat ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__Real__Oreal,type,
    map_upds_int_real: ( int > option_real ) > list_int > list_real > int > option_real ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up2395700066158073305_VEBTi: ( int > option_VEBT_VEBTi ) > list_int > list_VEBT_VEBTi > int > option_VEBT_VEBTi ).

thf(sy_c_Map_Omap__upds_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    map_up3480207007320876304T_VEBT: ( int > option_VEBT_VEBT ) > list_int > list_VEBT_VEBT > int > option_VEBT_VEBT ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001_Eo,type,
    map_upds_nat_o: ( nat > option_o ) > list_nat > list_o > nat > option_o ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Int__Oint,type,
    map_upds_nat_int: ( nat > option_int ) > list_nat > list_int > nat > option_int ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Nat__Onat,type,
    map_upds_nat_nat: ( nat > option_nat ) > list_nat > list_nat > nat > option_nat ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Num__Onum,type,
    map_upds_nat_num: ( nat > option_num ) > list_nat > list_num > nat > option_num ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_up1590851270032768825at_nat: ( nat > option4927543243414619207at_nat ) > list_nat > list_P6011104703257516679at_nat > nat > option4927543243414619207at_nat ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__Real__Oreal,type,
    map_upds_nat_real: ( nat > option_real ) > list_nat > list_real > nat > option_real ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up547622734759332733_VEBTi: ( nat > option_VEBT_VEBTi ) > list_nat > list_VEBT_VEBTi > nat > option_VEBT_VEBTi ).

thf(sy_c_Map_Omap__upds_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    map_up750602437722059884T_VEBT: ( nat > option_VEBT_VEBT ) > list_nat > list_VEBT_VEBT > nat > option_VEBT_VEBT ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001_Eo,type,
    map_upds_real_o: ( real > option_o ) > list_real > list_o > real > option_o ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Int__Oint,type,
    map_upds_real_int: ( real > option_int ) > list_real > list_int > real > option_int ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Nat__Onat,type,
    map_upds_real_nat: ( real > option_nat ) > list_real > list_nat > real > option_nat ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Num__Onum,type,
    map_upds_real_num: ( real > option_num ) > list_real > list_num > real > option_num ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__Real__Oreal,type,
    map_upds_real_real: ( real > option_real ) > list_real > list_real > real > option_real ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_up4277394084060657369_VEBTi: ( real > option_VEBT_VEBTi ) > list_real > list_VEBT_VEBTi > real > option_VEBT_VEBTi ).

thf(sy_c_Map_Omap__upds_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_up3774473461586936848T_VEBT: ( real > option_VEBT_VEBT ) > list_real > list_VEBT_VEBT > real > option_VEBT_VEBT ).

thf(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    map_upds_VEBT_VEBT_o: ( vEBT_VEBT > option_o ) > list_VEBT_VEBT > list_o > vEBT_VEBT > option_o ).

thf(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    map_up886849737537031018BT_int: ( vEBT_VEBT > option_int ) > list_VEBT_VEBT > list_int > vEBT_VEBT > option_int ).

thf(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_up889340208046081294BT_nat: ( vEBT_VEBT > option_nat ) > list_VEBT_VEBT > list_nat > vEBT_VEBT > option_nat ).

thf(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
    map_up6670044031056635736BT_num: ( vEBT_VEBT > option_num ) > list_VEBT_VEBT > list_num > vEBT_VEBT > option_num ).

thf(sy_c_Map_Omap__upds_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_up4960461728960030954T_real: ( vEBT_VEBT > option_real ) > list_VEBT_VEBT > list_real > vEBT_VEBT > option_real ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001_Eo,type,
    ran_nat_o: ( nat > option_o ) > set_o ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Int__Oint,type,
    ran_nat_int: ( nat > option_int ) > set_int ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Nat__Onat,type,
    ran_nat_nat: ( nat > option_nat ) > set_nat ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Num__Onum,type,
    ran_nat_num: ( nat > option_num ) > set_num ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ran_na2114640787166747904at_nat: ( nat > option4927543243414619207at_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Real__Oreal,type,
    ran_nat_real: ( nat > option_real ) > set_real ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    ran_nat_VEBT_VEBTi: ( nat > option_VEBT_VEBTi ) > set_VEBT_VEBTi ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    ran_nat_VEBT_VEBT: ( nat > option_VEBT_VEBT ) > set_VEBT_VEBT ).

thf(sy_c_Map_Orestrict__map_001_Eo_001_Eo,type,
    restrict_map_o_o: ( $o > option_o ) > set_o > $o > option_o ).

thf(sy_c_Map_Orestrict__map_001_Eo_001t__Nat__Onat,type,
    restrict_map_o_nat: ( $o > option_nat ) > set_o > $o > option_nat ).

thf(sy_c_Map_Orestrict__map_001_Eo_001t__Num__Onum,type,
    restrict_map_o_num: ( $o > option_num ) > set_o > $o > option_num ).

thf(sy_c_Map_Orestrict__map_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    restri2754309951621987225at_nat: ( $o > option4927543243414619207at_nat ) > set_o > $o > option4927543243414619207at_nat ).

thf(sy_c_Map_Orestrict__map_001_Eo_001t__Real__Oreal,type,
    restrict_map_o_real: ( $o > option_real ) > set_o > $o > option_real ).

thf(sy_c_Map_Orestrict__map_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    restri7010877615957751788nteger: ( code_integer > option_Code_integer ) > set_Code_integer > code_integer > option_Code_integer ).

thf(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Int__Oint,type,
    restrict_map_int_int: ( int > option_int ) > set_int > int > option_int ).

thf(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Nat__Onat,type,
    restrict_map_int_nat: ( int > option_nat ) > set_int > int > option_nat ).

thf(sy_c_Map_Orestrict__map_001t__Int__Oint_001t__Num__Onum,type,
    restrict_map_int_num: ( int > option_num ) > set_int > int > option_num ).

thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Nat__Onat,type,
    restrict_map_nat_nat: ( nat > option_nat ) > set_nat > nat > option_nat ).

thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Num__Onum,type,
    restrict_map_nat_num: ( nat > option_num ) > set_nat > nat > option_num ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001_Eo,type,
    restrict_map_real_o: ( real > option_o ) > set_real > real > option_o ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Int__Oint,type,
    restri6824647453968888714al_int: ( real > option_int ) > set_real > real > option_int ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Nat__Onat,type,
    restri6827137924477938990al_nat: ( real > option_nat ) > set_real > real > option_nat ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Num__Onum,type,
    restri3384469710633717624al_num: ( real > option_num ) > set_real > real > option_num ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    restri4688977514882730531at_nat: ( real > option4927543243414619207at_nat ) > set_real > real > option4927543243414619207at_nat ).

thf(sy_c_Map_Orestrict__map_001t__Real__Oreal_001t__Real__Oreal,type,
    restri4420043737358336266l_real: ( real > option_real ) > set_real > real > option_real ).

thf(sy_c_Map_Orestrict__map_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    restri1785250428630319752at_nat: ( set_nat > option_nat ) > set_set_nat > set_nat > option_nat ).

thf(sy_c_Map_Orestrict__map_001t__Set__Oset_It__Nat__Onat_J_001t__Num__Onum,type,
    restri7565954251640874194at_num: ( set_nat > option_num ) > set_set_nat > set_nat > option_num ).

thf(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    restri35548291844928020VEBT_o: ( vEBT_VEBT > option_o ) > set_VEBT_VEBT > vEBT_VEBT > option_o ).

thf(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    restri772377253954411184BT_int: ( vEBT_VEBT > option_int ) > set_VEBT_VEBT > vEBT_VEBT > option_int ).

thf(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    restri774867724463461460BT_nat: ( vEBT_VEBT > option_nat ) > set_VEBT_VEBT > vEBT_VEBT > option_nat ).

thf(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
    restri6555571547474015902BT_num: ( vEBT_VEBT > option_num ) > set_VEBT_VEBT > vEBT_VEBT > option_num ).

thf(sy_c_Map_Orestrict__map_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    restri7804378014469731632T_real: ( vEBT_VEBT > option_real ) > set_VEBT_VEBT > vEBT_VEBT > option_real ).

thf(sy_c_Misc_Omerge_001_Eo,type,
    merge_o: list_o > list_o > list_o ).

thf(sy_c_Misc_Omerge_001t__Int__Oint,type,
    merge_int: list_int > list_int > list_int ).

thf(sy_c_Misc_Omerge_001t__Nat__Onat,type,
    merge_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Misc_Omerge_001t__Num__Onum,type,
    merge_num: list_num > list_num > list_num ).

thf(sy_c_Misc_Omerge_001t__Rat__Orat,type,
    merge_rat: list_rat > list_rat > list_rat ).

thf(sy_c_Misc_Omerge_001t__Real__Oreal,type,
    merge_real: list_real > list_real > list_real ).

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_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__Int__Oint_J_J,type,
    size_s533118279054570080st_int: list_list_int > 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__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__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > nat ).

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > nat ).

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

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

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

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

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

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

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

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

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

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

thf(sy_c_NthRoot_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__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: complex > complex ).

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001_Eo,type,
    none_o: option_o ).

thf(sy_c_Option_Ooption_ONone_001t__Code____Numeral__Ointeger,type,
    none_Code_integer: option_Code_integer ).

thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    none_P4506660739021792380nteger: option2651255830984564193nteger ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    none_P2898468544434266898BT_nat: option8994051521412888707BT_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
    none_real: option_real ).

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_ONone_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Option_Ooption_OSome_001_Eo,type,
    some_o: $o > option_o ).

thf(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

thf(sy_c_Option_Ooption_OSome_001t__Code____Numeral__Ointeger,type,
    some_Code_integer: code_integer > option_Code_integer ).

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,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

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

thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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thf(sy_c_Option_Othese_001_Eo,type,
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thf(sy_c_Option_Othese_001t__Int__Oint,type,
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thf(sy_c_Option_Othese_001t__Nat__Onat,type,
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thf(sy_c_Option_Othese_001t__Num__Onum,type,
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thf(sy_c_Option_Othese_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Othese_001t__Real__Oreal,type,
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thf(sy_c_Option_Othese_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Othese_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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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,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    bot_bot_set_set_real: set_set_real ).

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_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__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $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__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    ord_le3602516367967493612eger_o: ( code_integer > code_integer > $o ) > ( code_integer > code_integer > $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__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__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__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le1182472622972956176_nat_o: ( vEBT_VEBT > nat > $o ) > ( vEBT_VEBT > nat > $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__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__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $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_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    ord_le3725938330318615451nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    ord_le3442269383143156041BT_nat: set_Pr7556676689462069481BT_nat > set_Pr7556676689462069481BT_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    ord_le1914454125413604393nteger: set_set_Code_integer > set_set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
    ord_le4750530260501030778omplex: set_set_complex > set_set_complex > $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__Real__Oreal_J_J,type,
    ord_le3558479182127378552t_real: set_set_real > set_set_real > $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__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_Omax_001_Eo,type,
    ord_max_o: $o > $o > $o ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_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_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: set_real > set_real > set_real ).

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

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

thf(sy_c_Orderings_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__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__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_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).

thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

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

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

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

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

thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
    product_Pair_o_real: $o > real > product_prod_o_real ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

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__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    produc101793102246108661omplex: complex > complex > produc4411394909380815293omplex ).

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__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__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__Num__Onum,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).

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__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
    product_Pair_real_o: real > $o > product_prod_real_o ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
    produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

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__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    produc8194178580519725514EBTi_o: vEBT_VEBTi > $o > produc5014006835512566296EBTi_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    produc7192665754729510430Ti_nat: vEBT_VEBTi > nat > produc3881548065746020326Ti_nat ).

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_Oprod_Ocase__prod_001_Eo_001_Eo_001_Eo,type,
    produc6197397395684419436_o_o_o: ( $o > $o > $o ) > product_prod_o_o > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001_Eo,type,
    produc9051730707245535732_nat_o: set_nat > set_o > set_Pr3149072824959771635_nat_o ).

thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    produc6734486746367016272t_real: set_nat > set_real > set_Pr320017278500174781t_real ).

thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc1441754205559175824T_VEBT: set_nat > set_VEBT_VEBT > set_Pr6167073792073659919T_VEBT ).

thf(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001_Eo,type,
    produc674561262062614936real_o: set_real > set_o > set_Pr4936984352647145239real_o ).

thf(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    produc2078423282641139152al_nat: set_real > set_nat > set_Pr3510011417693777981al_nat ).

thf(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    produc2998788613821421740l_real: set_real > set_real > set_Pr6218003697084177305l_real ).

thf(sy_c_Product__Type_Oproduct_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc7150050738623674420T_VEBT: set_real > set_VEBT_VEBT > set_Pr6019664923565264691T_VEBT ).

thf(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc2426872939560822710VEBT_o: set_VEBT_VEBT > set_o > set_Pr3175402225741728619VEBT_o ).

thf(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8336039005996768526T_real: set_VEBT_VEBT > set_real > set_Pr7765410600122031685T_real ).

thf(sy_c_Product__Type_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc3886929683002245970T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_Pr6192946355708809607T_VEBT ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Relation_OId__on_001_Eo,type,
    id_on_o: set_o > set_Product_prod_o_o ).

thf(sy_c_Relation_OId__on_001t__Code____Numeral__Ointeger,type,
    id_on_Code_integer: set_Code_integer > set_Pr4811707699266497531nteger ).

thf(sy_c_Relation_OId__on_001t__Complex__Ocomplex,type,
    id_on_complex: set_complex > set_Pr5085853215250843933omplex ).

thf(sy_c_Relation_OId__on_001t__Int__Oint,type,
    id_on_int: set_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
    id_on_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_OId__on_001t__Real__Oreal,type,
    id_on_real: set_real > set_Pr6218003697084177305l_real ).

thf(sy_c_Relation_OId__on_001t__VEBT____Definitions__OVEBT,type,
    id_on_VEBT_VEBT: set_VEBT_VEBT > set_Pr6192946355708809607T_VEBT ).

thf(sy_c_Relation_Ototal__on_001_Eo,type,
    total_on_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_Relation_Ototal__on_001t__Code____Numeral__Ointeger,type,
    total_8516208958838685657nteger: set_Code_integer > set_Pr4811707699266497531nteger > $o ).

thf(sy_c_Relation_Ototal__on_001t__Int__Oint,type,
    total_on_int: set_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
    total_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Ototal__on_001t__Real__Oreal,type,
    total_on_real: set_real > set_Pr6218003697084177305l_real > $o ).

thf(sy_c_Relation_Ototal__on_001t__VEBT____Definitions__OVEBT,type,
    total_on_VEBT_VEBT: set_VEBT_VEBT > set_Pr6192946355708809607T_VEBT > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set_OCollect_001_Eo,type,
    collect_o: ( $o > $o ) > set_o ).

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

thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

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

thf(sy_c_Set_OCollect_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    collec3483841146883906114nteger: ( list_Code_integer > $o ) > set_li6976499617229504675nteger ).

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

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

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    image_real_set_int: ( real > set_int ) > set_real > set_set_int ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
    image_real_set_nat: ( real > set_nat ) > set_real > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
    image_real_set_real: ( real > set_real ) > set_real > set_set_real ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    image_6925917818215209377T_VEBT: ( real > set_VEBT_VEBT ) > set_real > set_set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    image_real_VEBT_VEBT: ( real > vEBT_VEBT ) > set_real > set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
    image_set_complex_o: ( set_complex > $o ) > set_set_complex > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001_062_It__Int__Oint_M_Eo_J,type,
    image_set_int_int_o: ( set_int > int > $o ) > set_set_int > set_int_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    image_set_int_o: ( set_int > $o ) > set_set_int > set_o ).

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__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    image_set_list_nat_o: ( set_list_nat > $o ) > set_set_list_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
    image_set_nat_nat_o: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    image_2753855177019846445on_nat: ( set_nat > option_nat ) > set_set_nat > set_option_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Num__Onum_J,type,
    image_3462099113968684151on_num: ( set_nat > option_num ) > set_set_nat > set_option_num ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_838307146230912995et_nat: ( set_nat > option_set_nat ) > set_set_nat > set_option_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__Real__Oreal_J,type,
    image_6333053925516494319t_real: ( set_nat > set_real ) > set_set_nat > set_set_real ).

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__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    image_2931711902234744598eger_o: ( set_Pr4811707699266497531nteger > code_integer > code_integer > $o ) > set_se4344029326803248219nteger > set_Co5460287733524608044eger_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    image_4446226961036766042_int_o: ( set_Pr958786334691620121nt_int > int > int > $o ) > set_se6260736226359567993nt_int > set_int_int_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    image_6923511907955291850_nat_o: ( set_Pr1261947904930325089at_nat > nat > nat > $o ) > set_se7855581050983116737at_nat > set_nat_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    image_2055017250723459638_num_o: ( set_Pr6200539531224447659at_num > nat > num > $o ) > set_se4826145725398303499at_num > set_nat_num_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    image_4158226296049136288_nat_o: ( set_Pr7556676689462069481BT_nat > vEBT_VEBT > nat > $o ) > set_se3932177096832370463BT_nat > set_VEBT_VEBT_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_062_It__Real__Oreal_M_Eo_J,type,
    image_5650221686686655994real_o: ( set_real > real > $o ) > set_set_real > set_real_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_Eo,type,
    image_set_real_o: ( set_real > $o ) > set_set_real > set_o ).

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_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    image_4331731847045299910_nat_o: ( set_set_nat > set_nat > $o ) > set_set_set_nat > set_set_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
    image_set_set_nat_o: ( set_set_nat > $o ) > set_set_set_nat > set_o ).

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_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    image_7157618159374383128VEBT_o: ( set_VEBT_VEBT > vEBT_VEBT > $o ) > set_set_VEBT_VEBT > set_VEBT_VEBT_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001_Eo,type,
    image_5801891848100486793VEBT_o: ( set_VEBT_VEBT > $o ) > set_set_VEBT_VEBT > set_o ).

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__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__Option__Ooption_It__Nat__Onat_J,type,
    image_8844776943898047887on_nat: ( vEBT_VEBT > option_nat ) > set_VEBT_VEBT > set_option_nat ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Num__Onum_J,type,
    image_329648843992109785on_num: ( vEBT_VEBT > option_num ) > set_VEBT_VEBT > set_option_num ).

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__Set__Oset_I_Eo_J,type,
    image_7883550159813902793_set_o: ( vEBT_VEBT > set_o ) > set_VEBT_VEBT > set_set_o ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    image_7429379093677266050nteger: ( vEBT_VEBT > set_Code_integer ) > set_VEBT_VEBT > set_set_Code_integer ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    image_7711551513561399379omplex: ( vEBT_VEBT > set_complex ) > set_VEBT_VEBT > set_set_complex ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__Int__Oint_J,type,
    image_2273570491937255121et_int: ( vEBT_VEBT > set_int ) > set_VEBT_VEBT > set_set_int ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__Nat__Onat_J,type,
    image_6451421511446451829et_nat: ( vEBT_VEBT > set_nat ) > set_VEBT_VEBT > set_set_nat ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__Real__Oreal_J,type,
    image_6636839513470643793t_real: ( vEBT_VEBT > set_real ) > set_VEBT_VEBT > set_set_real ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    image_2685870239581809509T_VEBT: ( vEBT_VEBT > set_VEBT_VEBT ) > set_VEBT_VEBT > set_set_VEBT_VEBT ).

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__Extended____Nat__Oenat,type,
    insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).

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__Option__Ooption_I_Eo_J,type,
    insert_option_o: option_o > set_option_o > set_option_o ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Int__Oint_J,type,
    insert_option_int: option_int > set_option_int > set_option_int ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Nat__Onat_J,type,
    insert_option_nat: option_nat > set_option_nat > set_option_nat ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Num__Onum_J,type,
    insert_option_num: option_num > set_option_num > set_option_num ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    insert6842972033726296599at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > set_op4508134149509766951at_nat ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Real__Oreal_J,type,
    insert_option_real: option_real > set_option_real > set_option_real ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    insert6281397740311767046et_nat: option_set_nat > set_option_set_nat > set_option_set_nat ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
    insert9052096497370831178T_VEBT: option_VEBT_VEBT > set_option_VEBT_VEBT > set_option_VEBT_VEBT ).

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__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    insert4913895101485356395nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    insert3126710022685806477omplex: produc4411394909380815293omplex > set_Pr5085853215250843933omplex > set_Pr5085853215250843933omplex ).

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__Nat__Onat_Mt__Num__Onum_J,type,
    insert8920054152555992091at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    insert7746734233410687241l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > set_Pr6218003697084177305l_real ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    insert8978894354669351395BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > set_Pr7556676689462069481BT_nat ).

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__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__Int__Oint,type,
    is_singleton_int: set_int > $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__VEBT____Definitions__OVEBT,type,
    is_sin24926331636114728T_VEBT: set_VEBT_VEBT > $o ).

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__Real__Oreal,type,
    the_elem_real: set_real > real ).

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__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

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

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Num__Onum,type,
    set_fo8365102181078989356at_num: ( nat > num > num ) > nat > nat > num > num ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Num__Onum,type,
    set_fo256927282339908995el_num: produc3368934014287244435at_num > produc3368934014287244435at_num > $o ).

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_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_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_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: int > int > int ).

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

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

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

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

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

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > nat > $o ).

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

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

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

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

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

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

thf(sy_c_Time__Reasoning_Ofails_001_Eo,type,
    time_fails_o: heap_Time_Heap_o > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Time__Reasoning_Ofails_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_f8309030937046825446_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > $o ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Time__Reasoning_Otime_001t__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_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_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__Code____Numeral__Ointeger,type,
    transi6870300401645067644nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).

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__Real__Oreal,type,
    transi1789104906590519371l_real: set_Pr6218003697084177305l_real > set_Pr6218003697084177305l_real ).

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_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_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: nat > heap_Time_Heap_o > heap_T844314716496656296list_o ).

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

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

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

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

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__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_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o ) > ( $o > $o > heap_Time_Heap_o ) > vEBT_VEBTi > heap_Time_Heap_o ).

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

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

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__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__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_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).

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

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

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

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

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

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

thf(sy_c_VEBT__List__Assn_OlistI__assn_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_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__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__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: set_nat > ( real > nat > assn ) > list_real > list_nat > 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_001_Eo,type,
    vEBT_L7058566406413635588VEBT_o: set_nat > ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8648204552663881920BT_int: set_nat > ( vEBT_VEBT > int > assn ) > list_VEBT_VEBT > list_int > 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__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: set_nat > ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > 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_001_Eo,type,
    vEBT_L7363604446928714179sn_o_o: ( $o > $o > assn ) > list_o > list_o > assn ).

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

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

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L2956511777047245877on_nat: ( $o > option_nat > assn ) > list_o > list_option_nat > assn ).

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

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L3704169666673096010_VEBTi: ( $o > vEBT_VEBTi > assn ) > list_o > list_VEBT_VEBTi > 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__Nat__Onat_001_Eo,type,
    vEBT_L7887682484454631235_nat_o: ( nat > $o > assn ) > list_nat > list_o > assn ).

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

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

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

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

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

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

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

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > 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__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( vEBT_VEBT > int > assn ) > list_VEBT_VEBT > list_int > assn ).

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

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

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

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).

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

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

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
    accp_P4916641582247091100at_num: ( produc3368934014287244435at_num > produc3368934014287244435at_num > $o ) > produc3368934014287244435at_num > $o ).

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

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

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

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

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

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

thf(sy_c_Wellfounded_Omeasure_001t__Code____Numeral__Ointeger,type,
    measure_Code_integer: ( code_integer > nat ) > set_Pr4811707699266497531nteger ).

thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).

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__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > set_Extended_enat > $o ).

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

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

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

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

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

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

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

thf(sy_c_member_001t__Option__Ooption_It__Int__Oint_J,type,
    member_option_int: option_int > set_option_int > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
    member_option_nat: option_nat > set_option_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Num__Onum_J,type,
    member_option_num: option_num > set_option_num > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Real__Oreal_J,type,
    member_option_real: option_real > set_option_real > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    member8989860449721436141et_nat: option_set_nat > set_option_set_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
    member2458453091852628771T_VEBT: option_VEBT_VEBT > set_option_VEBT_VEBT > $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_I_Eo_Mt__Real__Oreal_J,type,
    member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $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__Nat__Onat_J,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $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__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__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_na____,type,
    na: nat ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_v____,type,
    v: product_prod_nat_nat ).

thf(sy_v_vg____,type,
    vg: list_VEBT_VEBT ).

thf(sy_v_vh____,type,
    vh: vEBT_VEBT ).

thf(sy_v_vi____,type,
    vi: nat ).

% Relevant facts (10206)
thf(fact_0_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_2_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_3_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S ) ) ) ).

% deg_SUcn_Node
thf(fact_4__C5_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ v ) @ ( suc @ zero_zero_nat ) @ vg @ vh ) @ na ).

% "5"
thf(fact_5_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_6_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_7_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_8_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_9_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_10_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_11_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_12_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_13_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_14_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_15_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_16_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_17_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_18_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_19_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_20_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_21_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_22_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_23_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_24_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_25_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_26_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_27_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_28_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_29_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_30_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_31_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_32_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_33_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_34_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_35_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_36_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_37_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_38_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_39_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_40_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_41_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X: option_nat] : ( times_times_assn @ ( Q @ X ) @ R ) ) ) ).

% frame_rule
thf(fact_42_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X: $o] : ( times_times_assn @ ( Q @ X ) @ R ) ) ) ).

% frame_rule
thf(fact_43_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X ) @ R ) ) ) ).

% frame_rule
thf(fact_44_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X: nat] : ( times_times_assn @ ( Q @ X ) @ R ) ) ) ).

% frame_rule
thf(fact_45_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ R @ P ) @ C
        @ ^ [X: option_nat] : ( times_times_assn @ R @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_46_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ R @ P ) @ C
        @ ^ [X: $o] : ( times_times_assn @ R @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_47_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ R @ P ) @ C
        @ ^ [X: vEBT_VEBTi] : ( times_times_assn @ R @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_48_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ R @ P ) @ C
        @ ^ [X: nat] : ( times_times_assn @ R @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_49_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R ) ) ) ).

% assn_times_assoc
thf(fact_50_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P2: assn,Q2: assn] : ( times_times_assn @ Q2 @ P2 ) ) ) ).

% assn_times_comm
thf(fact_51_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% star_assoc
thf(fact_52_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% star_aci(2)
thf(fact_53_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ A @ ( times_times_assn @ B @ C ) )
      = ( times_times_assn @ B @ ( times_times_assn @ A @ C ) ) ) ).

% star_aci(3)
thf(fact_54_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).

% assn_aci(10)
thf(fact_55_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_56_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_57_is__hoare__triple,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_58_is__hoare__triple,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ).

% is_hoare_triple
thf(fact_59_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,G: heap_T2636463487746394924on_nat] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_7629718768684598413on_nat @ P @ G @ Q ) )
       => ( hoare_7629718768684598413on_nat @ P @ ( if_Hea5867803462524415986on_nat @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_60_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn,G: heap_Time_Heap_o] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_hoare_triple_o @ P @ G @ Q ) )
       => ( hoare_hoare_triple_o @ P @ ( if_Heap_Time_Heap_o @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_61_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,G: heap_T8145700208782473153_VEBTi] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_1429296392585015714_VEBTi @ P @ G @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( if_Hea8453224502484754311_VEBTi @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_62_if__rule,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn,G: heap_Time_Heap_nat] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) )
     => ( ( ~ B
         => ( hoare_3067605981109127869le_nat @ P @ G @ Q ) )
       => ( hoare_3067605981109127869le_nat @ P @ ( if_Hea2662716070787841314ap_nat @ B @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_63_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_64_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_65_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_66_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_67_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_68_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_69_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_70_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_71_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_72_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_73_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_74_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_75_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_76_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_77_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( times_times_complex @ A @ B )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_78_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_79_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_80_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_81_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_82_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_83_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_84_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_85_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_86_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_87_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_88_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_89_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_90_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_91_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_92_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_93_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_94_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_95_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X2 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_96_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_97_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_98_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_99_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_100_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_101_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_102_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_103_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_104_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_105_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_106_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A3: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A3: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_119_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_120_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_121_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_122_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_123_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_124_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_125_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_126_zero__reorient,axiom,
    ! [X4: complex] :
      ( ( zero_zero_complex = X4 )
      = ( X4 = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_127_zero__reorient,axiom,
    ! [X4: real] :
      ( ( zero_zero_real = X4 )
      = ( X4 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_128_zero__reorient,axiom,
    ! [X4: rat] :
      ( ( zero_zero_rat = X4 )
      = ( X4 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_129_zero__reorient,axiom,
    ! [X4: nat] :
      ( ( zero_zero_nat = X4 )
      = ( X4 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_130_zero__reorient,axiom,
    ! [X4: int] :
      ( ( zero_zero_int = X4 )
      = ( X4 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_131_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_132_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_133_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_134_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_135_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% mult.assoc
thf(fact_136_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.assoc
thf(fact_137_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_138_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_139_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_140_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_141_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_142_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_143_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_144_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_145_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_147_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( times_times_assn @ B @ ( times_times_assn @ A @ C ) )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_148_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_149_Suc__inject,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( suc @ X4 )
        = ( suc @ Y ) )
     => ( X4 = Y ) ) ).

% Suc_inject
thf(fact_150_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_151_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_152_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_153_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_154_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_155_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_156_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_157_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_158_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_159_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_160_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_161_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_162_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_163_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_164_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_165_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X4 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X4 @ Sx ) ) ) ).

% succ_correct
thf(fact_166_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_167_list__decode_Ocases,axiom,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X4
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_168_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,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ( ( X4
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X4
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_169_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_170_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_171_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_172_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_173_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_174_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_175_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_176_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_177_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_178_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_179_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_180_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_181_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_182_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_183_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_184_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_185_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_186_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_187_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_188_linorder__neqE__nat,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_nat @ X4 @ Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neqE_nat
thf(fact_189_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_190_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_191_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_192_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_193_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_194_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_195_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_196_linorder__neqE__linordered__idom,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_197_linorder__neqE__linordered__idom,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_rat @ X4 @ Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_198_linorder__neqE__linordered__idom,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_int @ X4 @ Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_size__neq__size__imp__neq,axiom,
    ! [X4: list_real,Y: list_real] :
      ( ( ( size_size_list_real @ X4 )
       != ( size_size_list_real @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_210_size__neq__size__imp__neq,axiom,
    ! [X4: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X4 )
       != ( size_size_list_o @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_211_size__neq__size__imp__neq,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X4 )
       != ( size_size_list_nat @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_212_size__neq__size__imp__neq,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X4 )
       != ( size_size_list_int @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_213_size__neq__size__imp__neq,axiom,
    ! [X4: num,Y: num] :
      ( ( ( size_size_num @ X4 )
       != ( size_size_num @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_214_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_215_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_216_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_217_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_218_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_219_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_220_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_221_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_222_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_223_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_224_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_225_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_226_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_227_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_228_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_229_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_230_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_231_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_232_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_233_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_234_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_235_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_236_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_237_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_238_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_239_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_240_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_241_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_242_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_243_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_244_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_245_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_246_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_247_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_248_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_249_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_250_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_251_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_252_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_253_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_254_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_255_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_256_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_257_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_258_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_259_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_260_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_261_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_262_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_263_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_264_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_265_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_266_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_267_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_268_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_269_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_270_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_271_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_272_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_273_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_274_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_275_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_307_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_308_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_309_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_310_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_311_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_312_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_313_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% gr0_implies_Suc
thf(fact_314_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_315_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_316_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X4 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Sx ) ) ) ).

% succ_corr
thf(fact_325_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_326_less__option__Some,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_option_real @ ( some_real @ X4 ) @ ( some_real @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% less_option_Some
thf(fact_327_less__option__Some,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X4 ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X4 @ Y ) ) ).

% less_option_Some
thf(fact_328_less__option__Some,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X4 ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X4 @ Y ) ) ).

% less_option_Some
thf(fact_329_less__option__Some,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X4 @ Y ) ) ).

% less_option_Some
thf(fact_330_less__option__Some,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X4 ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X4 @ Y ) ) ).

% less_option_Some
thf(fact_331_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y4: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y4 ) ) ) ) ).

% greater_shift
thf(fact_332_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_333_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_334_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y4 ) ) ) ) ).

% less_shift
thf(fact_335_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X4: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X4 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ X4 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_336_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ zero_zero_nat ) )
           => ( P @ I3 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_337_forall__finite_I3_J,axiom,
    ! [X4: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ X4 ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X4 ) )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_338_mult__less__iff1,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_339_mult__less__iff1,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_340_mult__less__iff1,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_341_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X4 )
        = ( member_nat @ X4 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_342_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_343_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X4 )
      @ ^ [R2: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_member @ T @ X4 ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_344_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_345_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_346_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_347_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_348_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_349_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I4: nat] :
      ( ( ord_less_nat @ I4 @ zero_zero_nat )
     => ( P @ I4 ) ) ).

% forall_finite(1)
thf(fact_350_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 ) ).

% vebt_member.simps(4)
thf(fact_351_succ__member,axiom,
    ! [T: vEBT_VEBT,X4: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X4 @ Y )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X4 @ Z2 ) )
           => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ).

% succ_member
thf(fact_352_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X4: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X4 ) ).

% vebt_member.simps(3)
thf(fact_353_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X4: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).

% buildup_nothing_in_leaf
thf(fact_354_mul__shift,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X4 @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_355_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_356_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_357_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X4: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X4 ) ).

% buildup_nothing_in_min_max
thf(fact_358_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs )
             => ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_359_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs )
             => ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ).

% min_in_set_def
thf(fact_360_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X4 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X4 )
          | ( vEBT_VEBT_membermima @ Tree @ X4 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_361_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X4 )
         => ( ord_less_eq_nat @ X4 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_362_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_363_less__eq__option__Some,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ X4 ) @ ( some_set_int @ Y ) )
      = ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% less_eq_option_Some
thf(fact_364_less__eq__option__Some,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X4 ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X4 @ Y ) ) ).

% less_eq_option_Some
thf(fact_365_less__eq__option__Some,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X4 ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X4 @ Y ) ) ).

% less_eq_option_Some
thf(fact_366_less__eq__option__Some,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X4 @ Y ) ) ).

% less_eq_option_Some
thf(fact_367_less__eq__option__Some,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X4 ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X4 @ Y ) ) ).

% less_eq_option_Some
thf(fact_368_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_369_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_370_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_371_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X4 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).

% maxt_corr
thf(fact_372_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X4 ) ) ) ) ).

% maxt_sound
thf(fact_373_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_374_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_375_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_376_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y4 ) ) ) ) ).

% lesseq_shift
thf(fact_377_pred__member,axiom,
    ! [T: vEBT_VEBT,X4: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X4 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X4 ) )
           => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ).

% pred_member
thf(fact_378_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_379_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_380_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_381_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_382_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_383_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_384_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_zero__le,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).

% zero_le
thf(fact_396_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_397_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_398_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_399_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_400_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_401_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_402_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M2: nat] :
          ( M6
          = ( suc @ M2 ) ) ) ).

% Suc_le_D
thf(fact_403_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_404_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_405_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_406_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_407_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_408_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_409_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_410_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_411_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_412_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_413_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
          | ( M5 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_414_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_415_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
          & ( M5 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_416_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_417_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_418_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_419_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_420_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_421_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_422_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_423_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_424_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_425_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_453_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_454_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_455_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_456_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_457_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_458_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_459_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_460_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_461_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_462_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_463_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_464_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_465_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_466_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_467_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_468_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_469_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_470_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_471_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_472_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_473_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_474_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_475_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_476_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_477_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_478_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_479_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_480_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_481_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_482_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_483_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_484_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_485_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_486_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_487_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_488_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_489_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_490_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_491_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_492_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_493_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_494_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_495_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_496_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_497_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_498_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_499_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_500_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_501_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_502_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_503_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_504_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_505_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_506_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_507_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_508_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_509_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_510_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_511_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_512_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_513_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_514_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_515_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_516_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_517_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_518_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_519_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_520_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_521_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_522_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_523_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_524_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_525_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_526_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_527_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_528_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_529_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_530_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_531_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_532_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_533_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_534_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_535_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_536_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_537_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_538_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_539_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_540_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_541_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_542_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_543_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_544_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_545_mult__le__cancel__iff2,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_546_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X4 ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_547_mult__le__cancel__iff2,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_548_mult__le__cancel__iff1,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_549_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_550_mult__le__cancel__iff1,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_551_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_552_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_553_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R2: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_554_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X4 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ X4 @ Y4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_555_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_556_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs )
          & ( ord_less_nat @ X @ Y4 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ X @ Z2 )
               => ( ord_less_eq_nat @ Y4 @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_557_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_558_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_559_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_560_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_561_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X4 )
         => ( ord_less_eq_nat @ Mini @ X4 ) ) ) ) ).

% mint_corr_help
thf(fact_562_empty__iff,axiom,
    ! [C: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ).

% empty_iff
thf(fact_563_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_564_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_565_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_566_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_567_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_568_all__not__in__conv,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( ! [X: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X @ A3 ) )
      = ( A3 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
thf(fact_569_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ! [X: set_nat] :
            ~ ( member_set_nat @ X @ A3 ) )
      = ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_570_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ! [X: real] :
            ~ ( member_real @ X @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_571_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o @ X @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_572_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_573_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ! [X: int] :
            ~ ( member_int @ X @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_574_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_575_empty__subsetI,axiom,
    ! [A3: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A3 ) ).

% empty_subsetI
thf(fact_576_empty__subsetI,axiom,
    ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).

% empty_subsetI
thf(fact_577_empty__subsetI,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).

% empty_subsetI
thf(fact_578_empty__subsetI,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).

% empty_subsetI
thf(fact_579_subset__empty,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
      = ( A3 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_580_subset__empty,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_581_subset__empty,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_582_subset__empty,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_583_empty__Collect__eq,axiom,
    ! [P: complex > $o] :
      ( ( bot_bot_set_complex
        = ( collect_complex @ P ) )
      = ( ! [X: complex] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_584_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_585_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_586_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_587_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_588_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_589_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_590_Collect__empty__eq,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( ! [X: complex] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_591_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_592_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_593_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_594_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_595_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_596_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_597_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_598_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X4 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 ) ) ) ).

% mint_corr
thf(fact_599_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X4 ) ) ) ) ).

% mint_sound
thf(fact_600_not__None__eq,axiom,
    ! [X4: option4927543243414619207at_nat] :
      ( ( X4 != none_P5556105721700978146at_nat )
      = ( ? [Y4: product_prod_nat_nat] :
            ( X4
            = ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_601_not__None__eq,axiom,
    ! [X4: option_nat] :
      ( ( X4 != none_nat )
      = ( ? [Y4: nat] :
            ( X4
            = ( some_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_602_not__None__eq,axiom,
    ! [X4: option_num] :
      ( ( X4 != none_num )
      = ( ? [Y4: num] :
            ( X4
            = ( some_num @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_603_not__Some__eq,axiom,
    ! [X4: option4927543243414619207at_nat] :
      ( ( ! [Y4: product_prod_nat_nat] :
            ( X4
           != ( some_P7363390416028606310at_nat @ Y4 ) ) )
      = ( X4 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_604_not__Some__eq,axiom,
    ! [X4: option_nat] :
      ( ( ! [Y4: nat] :
            ( X4
           != ( some_nat @ Y4 ) ) )
      = ( X4 = none_nat ) ) ).

% not_Some_eq
thf(fact_605_not__Some__eq,axiom,
    ! [X4: option_num] :
      ( ( ! [Y4: num] :
            ( X4
           != ( some_num @ Y4 ) ) )
      = ( X4 = none_num ) ) ).

% not_Some_eq
thf(fact_606_less__eq__option__None__code,axiom,
    ! [X4: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X4 ) ).

% less_eq_option_None_code
thf(fact_607_less__eq__option__None__code,axiom,
    ! [X4: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X4 ) ).

% less_eq_option_None_code
thf(fact_608_less__option__None,axiom,
    ! [X4: option_nat] :
      ~ ( ord_less_option_nat @ X4 @ none_nat ) ).

% less_option_None
thf(fact_609_less__option__None,axiom,
    ! [X4: option_num] :
      ~ ( ord_less_option_num @ X4 @ none_num ) ).

% less_option_None
thf(fact_610_less__eq__option__Some__None,axiom,
    ! [X4: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X4 ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_611_less__eq__option__Some__None,axiom,
    ! [X4: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X4 ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_612_less__option__None__Some__code,axiom,
    ! [X4: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X4 ) ) ).

% less_option_None_Some_code
thf(fact_613_less__option__None__Some__code,axiom,
    ! [X4: num] : ( ord_less_option_num @ none_num @ ( some_num @ X4 ) ) ).

% less_option_None_Some_code
thf(fact_614_less__eq__option__None__is__None,axiom,
    ! [X4: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X4 @ none_nat )
     => ( X4 = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_615_less__eq__option__None__is__None,axiom,
    ! [X4: option_num] :
      ( ( ord_le6622620407824499402on_num @ X4 @ none_num )
     => ( X4 = none_num ) ) ).

% less_eq_option_None_is_None
thf(fact_616_less__eq__option__None,axiom,
    ! [X4: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X4 ) ).

% less_eq_option_None
thf(fact_617_less__eq__option__None,axiom,
    ! [X4: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X4 ) ).

% less_eq_option_None
thf(fact_618_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_619_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_620_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_621_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_622_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_623_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_624_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_625_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_626_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_627_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X5: option4927543243414619207at_nat] : ( P3 @ X5 ) )
    = ( ^ [P2: option4927543243414619207at_nat > $o] :
          ( ( P2 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P2 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_628_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X5: option_nat] : ( P3 @ X5 ) )
    = ( ^ [P2: option_nat > $o] :
          ( ( P2 @ none_nat )
          | ? [X: nat] : ( P2 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_629_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X5: option_num] : ( P3 @ X5 ) )
    = ( ^ [P2: option_num > $o] :
          ( ( P2 @ none_num )
          | ? [X: num] : ( P2 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_630_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X5: option4927543243414619207at_nat] : ( P3 @ X5 ) )
    = ( ^ [P2: option4927543243414619207at_nat > $o] :
          ( ( P2 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P2 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_631_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X5: option_nat] : ( P3 @ X5 ) )
    = ( ^ [P2: option_nat > $o] :
          ( ( P2 @ none_nat )
          & ! [X: nat] : ( P2 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_632_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X5: option_num] : ( P3 @ X5 ) )
    = ( ^ [P2: option_num > $o] :
          ( ( P2 @ none_num )
          & ! [X: num] : ( P2 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_633_combine__options__cases,axiom,
    ! [X4: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X4 = none_P5556105721700978146at_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X4
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_634_combine__options__cases,axiom,
    ! [X4: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X4 = none_P5556105721700978146at_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: product_prod_nat_nat,B3: nat] :
              ( ( X4
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_635_combine__options__cases,axiom,
    ! [X4: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X4 = none_P5556105721700978146at_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: product_prod_nat_nat,B3: num] :
              ( ( X4
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_636_combine__options__cases,axiom,
    ! [X4: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X4 = none_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: nat,B3: product_prod_nat_nat] :
              ( ( X4
                = ( some_nat @ A4 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_637_combine__options__cases,axiom,
    ! [X4: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X4 = none_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: nat,B3: nat] :
              ( ( X4
                = ( some_nat @ A4 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_638_combine__options__cases,axiom,
    ! [X4: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X4 = none_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: nat,B3: num] :
              ( ( X4
                = ( some_nat @ A4 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_639_combine__options__cases,axiom,
    ! [X4: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X4 = none_num )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: num,B3: product_prod_nat_nat] :
              ( ( X4
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_640_combine__options__cases,axiom,
    ! [X4: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X4 = none_num )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: num,B3: nat] :
              ( ( X4
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_641_combine__options__cases,axiom,
    ! [X4: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X4 = none_num )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X4 @ Y ) )
       => ( ! [A4: num,B3: num] :
              ( ( X4
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_642_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_643_less__option__None__is__Some,axiom,
    ! [X4: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X4 )
     => ? [Z3: nat] :
          ( X4
          = ( some_nat @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_644_less__option__None__is__Some,axiom,
    ! [X4: option_num] :
      ( ( ord_less_option_num @ none_num @ X4 )
     => ? [Z3: num] :
          ( X4
          = ( some_num @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_645_less__option__None__Some,axiom,
    ! [X4: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X4 ) ) ).

% less_option_None_Some
thf(fact_646_less__option__None__Some,axiom,
    ! [X4: num] : ( ord_less_option_num @ none_num @ ( some_num @ X4 ) ) ).

% less_option_None_Some
thf(fact_647_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_648_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_set_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_649_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_rat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_650_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_num @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_651_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_nat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_652_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_653_memb__imp__not__empty,axiom,
    ! [X4: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ S3 )
     => ( S3 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
thf(fact_654_memb__imp__not__empty,axiom,
    ! [X4: set_nat,S3: set_set_nat] :
      ( ( member_set_nat @ X4 @ S3 )
     => ( S3 != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_655_memb__imp__not__empty,axiom,
    ! [X4: real,S3: set_real] :
      ( ( member_real @ X4 @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
thf(fact_656_memb__imp__not__empty,axiom,
    ! [X4: $o,S3: set_o] :
      ( ( member_o @ X4 @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_657_memb__imp__not__empty,axiom,
    ! [X4: nat,S3: set_nat] :
      ( ( member_nat @ X4 @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_658_memb__imp__not__empty,axiom,
    ! [X4: int,S3: set_int] :
      ( ( member_int @ X4 @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_659_set__notEmptyE,axiom,
    ! [S3: set_VEBT_VEBT] :
      ( ( S3 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X3: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_660_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_661_set__notEmptyE,axiom,
    ! [S3: set_real] :
      ( ( S3 != bot_bot_set_real )
     => ~ ! [X3: real] :
            ~ ( member_real @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_662_set__notEmptyE,axiom,
    ! [S3: set_o] :
      ( ( S3 != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_663_set__notEmptyE,axiom,
    ! [S3: set_nat] :
      ( ( S3 != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_664_set__notEmptyE,axiom,
    ! [S3: set_int] :
      ( ( S3 != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_665_ex__in__conv,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( ? [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A3 ) )
      = ( A3 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
thf(fact_666_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ? [X: set_nat] : ( member_set_nat @ X @ A3 ) )
      = ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_667_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ? [X: real] : ( member_real @ X @ A3 ) )
      = ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_668_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X: $o] : ( member_o @ X @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_669_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_670_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ? [X: int] : ( member_int @ X @ A3 ) )
      = ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_671_equals0I,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ! [Y3: vEBT_VEBT] :
          ~ ( member_VEBT_VEBT @ Y3 @ A3 )
     => ( A3 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
thf(fact_672_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A3 )
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_673_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A3 )
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_674_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_675_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_676_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A3 )
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_677_equals0D,axiom,
    ! [A3: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A3 = bot_bo8194388402131092736T_VEBT )
     => ~ ( member_VEBT_VEBT @ A @ A3 ) ) ).

% equals0D
thf(fact_678_equals0D,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_679_equals0D,axiom,
    ! [A3: set_real,A: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ ( member_real @ A @ A3 ) ) ).

% equals0D
thf(fact_680_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o @ A @ A3 ) ) ).

% equals0D
thf(fact_681_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_682_equals0D,axiom,
    ! [A3: set_int,A: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ ( member_int @ A @ A3 ) ) ).

% equals0D
thf(fact_683_emptyE,axiom,
    ! [A: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ).

% emptyE
thf(fact_684_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_685_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_686_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_687_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_688_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_689_not__psubset__empty,axiom,
    ! [A3: set_real] :
      ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_690_not__psubset__empty,axiom,
    ! [A3: set_o] :
      ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_691_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_692_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_693_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_694_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_695_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_696_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_697_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $false ) ) ).

% Set.empty_def
thf(fact_698_Set_Oempty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X: list_nat] : $false ) ) ).

% Set.empty_def
thf(fact_699_Set_Oempty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X: set_nat] : $false ) ) ).

% Set.empty_def
thf(fact_700_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% Set.empty_def
thf(fact_701_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% Set.empty_def
thf(fact_702_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_703_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X: int] : $false ) ) ).

% Set.empty_def
thf(fact_704_le__some__optE,axiom,
    ! [M: set_int,X4: option_set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ M ) @ X4 )
     => ~ ! [M7: set_int] :
            ( ( X4
              = ( some_set_int @ M7 ) )
           => ~ ( ord_less_eq_set_int @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_705_le__some__optE,axiom,
    ! [M: rat,X4: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X4 )
     => ~ ! [M7: rat] :
            ( ( X4
              = ( some_rat @ M7 ) )
           => ~ ( ord_less_eq_rat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_706_le__some__optE,axiom,
    ! [M: num,X4: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X4 )
     => ~ ! [M7: num] :
            ( ( X4
              = ( some_num @ M7 ) )
           => ~ ( ord_less_eq_num @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_707_le__some__optE,axiom,
    ! [M: nat,X4: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X4 )
     => ~ ! [M7: nat] :
            ( ( X4
              = ( some_nat @ M7 ) )
           => ~ ( ord_less_eq_nat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_708_le__some__optE,axiom,
    ! [M: int,X4: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X4 )
     => ~ ! [M7: int] :
            ( ( X4
              = ( some_int @ M7 ) )
           => ~ ( ord_less_eq_int @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_709_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N5: nat] :
            ( ( ord_less_nat @ N5 @ N )
           => ~ ( P @ N5 ) )
       => ( P @ N ) )
     => ? [N6: nat] :
          ( ( ord_less_eq_nat @ N6 @ N )
          & ( P @ N6 ) ) ) ).

% exists_leI
thf(fact_710_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R2: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_711_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X4 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ Y4 @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_712_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X4 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X4 @ Sx ) ) ) ).

% pred_correct
thf(fact_713_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X4 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X4 @ Px ) ) ) ).

% pred_corr
thf(fact_714_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X4 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_715_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs )
          & ( ord_less_nat @ Y4 @ X )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ Z2 @ X )
               => ( ord_less_eq_nat @ Z2 @ Y4 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_716_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_717_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_718_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_719_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X4: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X4 @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ A4 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X4 @ A4 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_720_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X4: num > num > num,Xa: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X4 @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xa
                  = ( some_num @ A4 ) )
               => ! [B3: num] :
                    ( ( Xb
                      = ( some_num @ B3 ) )
                   => ( Y
                     != ( some_num @ ( X4 @ A4 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_721_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X4: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X4 @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xa
                  = ( some_nat @ A4 ) )
               => ! [B3: nat] :
                    ( ( Xb
                      = ( some_nat @ B3 ) )
                   => ( Y
                     != ( some_nat @ ( X4 @ A4 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_722_obtain__set__pred,axiom,
    ! [Z: nat,X4: nat,A3: set_nat] :
      ( ( ord_less_nat @ Z @ X4 )
     => ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
       => ( ( finite_finite_nat @ A3 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A3 @ X4 @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_723_obtain__set__succ,axiom,
    ! [X4: nat,Z: nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_nat @ X4 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
       => ( ( finite_finite_nat @ B4 )
         => ( ( A3 = B4 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A3 @ X4 @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_724_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_725_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ Xs2 )
              & ( ord_less_nat @ A @ X6 ) ) ) ) ).

% succ_none_empty
thf(fact_726_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ Xs2 )
              & ( ord_less_nat @ X6 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_727_subsetI,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_nat @ X3 @ B4 ) )
     => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).

% subsetI
thf(fact_728_subsetI,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( member_VEBT_VEBT @ X3 @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ A3 @ B4 ) ) ).

% subsetI
thf(fact_729_subsetI,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_real @ X3 @ B4 ) )
     => ( ord_less_eq_set_real @ A3 @ B4 ) ) ).

% subsetI
thf(fact_730_subsetI,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( member_set_nat @ X3 @ B4 ) )
     => ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ).

% subsetI
thf(fact_731_subsetI,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_int @ X3 @ B4 ) )
     => ( ord_less_eq_set_int @ A3 @ B4 ) ) ).

% subsetI
thf(fact_732_subset__antisym,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ A3 )
       => ( A3 = B4 ) ) ) ).

% subset_antisym
thf(fact_733_not__Some__eq2,axiom,
    ! [V: option8994051521412888707BT_nat] :
      ( ( ! [X: vEBT_VEBT,Y4: nat] :
            ( V
           != ( some_P3304306041812697742BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) ) ) )
      = ( V = none_P2898468544434266898BT_nat ) ) ).

% not_Some_eq2
thf(fact_734_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ( ! [X: int,Y4: int] :
            ( V
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X @ Y4 ) ) ) )
      = ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_735_not__Some__eq2,axiom,
    ! [V: option2651255830984564193nteger] :
      ( ( ! [X: code_integer,Y4: code_integer] :
            ( V
           != ( some_P6772290148444788224nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) ) ) )
      = ( V = none_P4506660739021792380nteger ) ) ).

% not_Some_eq2
thf(fact_736_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ( ! [X: nat,Y4: num] :
            ( V
           != ( some_P8071634352977444016at_num @ ( product_Pair_nat_num @ X @ Y4 ) ) ) )
      = ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
thf(fact_737_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ( ! [X: nat,Y4: nat] :
            ( V
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) ) ) )
      = ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_738_psubsetI,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( A3 != B4 )
       => ( ord_less_set_int @ A3 @ B4 ) ) ) ).

% psubsetI
thf(fact_739_psubsetE,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A3 @ B4 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B4 )
         => ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ).

% psubsetE
thf(fact_740_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_741_psubset__imp__subset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ A3 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_742_psubset__subset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_set_int @ A3 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C2 )
       => ( ord_less_set_int @ A3 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_743_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_744_subset__psubset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( ord_less_set_int @ B4 @ C2 )
       => ( ord_less_set_int @ A3 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_745_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_set_int @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_746_in__mono,axiom,
    ! [A3: set_nat,B4: set_nat,X4: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( member_nat @ X4 @ A3 )
       => ( member_nat @ X4 @ B4 ) ) ) ).

% in_mono
thf(fact_747_in__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( member_VEBT_VEBT @ X4 @ B4 ) ) ) ).

% in_mono
thf(fact_748_in__mono,axiom,
    ! [A3: set_real,B4: set_real,X4: real] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ( member_real @ X4 @ A3 )
       => ( member_real @ X4 @ B4 ) ) ) ).

% in_mono
thf(fact_749_in__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,X4: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ( member_set_nat @ X4 @ A3 )
       => ( member_set_nat @ X4 @ B4 ) ) ) ).

% in_mono
thf(fact_750_in__mono,axiom,
    ! [A3: set_int,B4: set_int,X4: int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( member_int @ X4 @ A3 )
       => ( member_int @ X4 @ B4 ) ) ) ).

% in_mono
thf(fact_751_subsetD,axiom,
    ! [A3: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_752_subsetD,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ( member_VEBT_VEBT @ C @ A3 )
       => ( member_VEBT_VEBT @ C @ B4 ) ) ) ).

% subsetD
thf(fact_753_subsetD,axiom,
    ! [A3: set_real,B4: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_754_subsetD,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_755_subsetD,axiom,
    ! [A3: set_int,B4: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B4 ) ) ) ).

% subsetD
thf(fact_756_equalityE,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B4 )
         => ~ ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ).

% equalityE
thf(fact_757_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A5 )
         => ( member_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_758_subset__eq,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
        ! [X: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X @ A5 )
         => ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_759_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A5 )
         => ( member_real @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_760_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
        ! [X: set_nat] :
          ( ( member_set_nat @ X @ A5 )
         => ( member_set_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_761_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A5 )
         => ( member_int @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_762_equalityD1,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => ( ord_less_eq_set_int @ A3 @ B4 ) ) ).

% equalityD1
thf(fact_763_Set_OequalityD2,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => ( ord_less_eq_set_int @ B4 @ A3 ) ) ).

% Set.equalityD2
thf(fact_764_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_765_subset__iff,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
        ! [T2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ T2 @ A5 )
         => ( member_VEBT_VEBT @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_766_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A5 )
         => ( member_real @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_767_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A5 )
         => ( member_set_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_768_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A5 )
         => ( member_int @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_769_subset__refl,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).

% subset_refl
thf(fact_770_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_771_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_772_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_773_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_774_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_775_subset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C2 )
       => ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).

% subset_trans
thf(fact_776_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_777_Collect__subset,axiom,
    ! [A3: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_778_Collect__subset,axiom,
    ! [A3: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_779_Collect__subset,axiom,
    ! [A3: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_780_Collect__subset,axiom,
    ! [A3: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_781_Collect__subset,axiom,
    ! [A3: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_782_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_783_Collect__subset,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_784_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A5 )
          @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_785_less__eq__set__def,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( ord_le418104280809901481VEBT_o
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A5 )
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_786_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A5 )
          @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_787_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_788_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A5 )
          @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_789_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_790_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_791_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_792_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_793_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_794_subset__Collect__conv,axiom,
    ! [S3: set_complex,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ S3 @ ( collect_complex @ P ) )
      = ( ! [X: complex] :
            ( ( member_complex @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_795_subset__Collect__conv,axiom,
    ! [S3: set_list_nat,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ S3 @ ( collect_list_nat @ P ) )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_796_subset__Collect__conv,axiom,
    ! [S3: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ S3 @ ( collect_set_nat @ P ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_797_subset__Collect__conv,axiom,
    ! [S3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S3 @ ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_798_subset__Collect__conv,axiom,
    ! [S3: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ S3 @ ( collect_int @ P ) )
      = ( ! [X: int] :
            ( ( member_int @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_799_prod__decode__aux_Ocases,axiom,
    ! [X4: product_prod_nat_nat] :
      ~ ! [K2: nat,M2: nat] :
          ( X4
         != ( product_Pair_nat_nat @ K2 @ M2 ) ) ).

% prod_decode_aux.cases
thf(fact_800_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_801_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_802_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_803_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X4 ) ).

% vebt_member.simps(2)
thf(fact_804_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_805_filter__preserves__multiset,axiom,
    ! [M8: list_nat > nat,P: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_806_filter__preserves__multiset,axiom,
    ! [M8: set_nat > nat,P: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_807_filter__preserves__multiset,axiom,
    ! [M8: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_808_filter__preserves__multiset,axiom,
    ! [M8: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_809_filter__preserves__multiset,axiom,
    ! [M8: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_810_filter__preserves__multiset,axiom,
    ! [M8: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_811_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_812_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_813_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_814_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_815_add__mset__in__multiset,axiom,
    ! [M8: list_nat > nat,A: list_nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_816_add__mset__in__multiset,axiom,
    ! [M8: set_nat > nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_817_add__mset__in__multiset,axiom,
    ! [M8: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_818_add__mset__in__multiset,axiom,
    ! [M8: int > nat,A: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_819_add__mset__in__multiset,axiom,
    ! [M8: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_820_add__mset__in__multiset,axiom,
    ! [M8: code_integer > nat,A: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_821_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_822_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_823_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_824_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X4 )
      = ( ( X4 = Mi )
        | ( X4 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_825_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_826_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_827_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_828_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_829_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_830_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_831_finite__Collect__subsets,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_832_finite__Collect__subsets,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B5: set_Code_integer] : ( ord_le7084787975880047091nteger @ B5 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_833_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_834_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X4: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] :
          ( X4
         != ( vEBT_Leaf @ A4 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X4
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_835_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_836_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_837_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_838_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_839_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_840_finite__Collect__conjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        | ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_841_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_842_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_843_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_844_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_845_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_846_finite__Collect__disjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        & ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_847_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,
    ! [X4: vEBT_VEBT] :
      ( ( X4
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X4
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X4
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X4
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X4
                 != ( 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_848_infinite__growing,axiom,
    ! [X7: set_Code_integer] :
      ( ( X7 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X7 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X7 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X7 ) ) ) ).

% infinite_growing
thf(fact_849_infinite__growing,axiom,
    ! [X7: set_o] :
      ( ( X7 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X7 )
           => ? [Xa2: $o] :
                ( ( member_o @ Xa2 @ X7 )
                & ( ord_less_o @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_o @ X7 ) ) ) ).

% infinite_growing
thf(fact_850_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X7 )
           => ? [Xa2: real] :
                ( ( member_real @ Xa2 @ X7 )
                & ( ord_less_real @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_real @ X7 ) ) ) ).

% infinite_growing
thf(fact_851_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X7 )
           => ? [Xa2: rat] :
                ( ( member_rat @ Xa2 @ X7 )
                & ( ord_less_rat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_rat @ X7 ) ) ) ).

% infinite_growing
thf(fact_852_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X7 )
           => ? [Xa2: num] :
                ( ( member_num @ Xa2 @ X7 )
                & ( ord_less_num @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_num @ X7 ) ) ) ).

% infinite_growing
thf(fact_853_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X7 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X7 ) ) ) ).

% infinite_growing
thf(fact_854_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X7 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X7 )
                & ( ord_less_int @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X7 ) ) ) ).

% infinite_growing
thf(fact_855_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S3 )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S3 )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_856_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ S3 )
            & ~ ? [Xa2: $o] :
                  ( ( member_o @ Xa2 @ S3 )
                  & ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_857_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S3 )
            & ~ ? [Xa2: real] :
                  ( ( member_real @ Xa2 @ S3 )
                  & ( ord_less_real @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_858_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S3 )
            & ~ ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ S3 )
                  & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_859_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S3 )
            & ~ ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ S3 )
                  & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_860_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S3 )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S3 )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_861_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S3 )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S3 )
                  & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_862_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X4: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X4
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X4
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X4
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_863_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X4: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X4
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X4
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A4: nat,B3: nat] :
              ( X4
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_864_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X4: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X4
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X4
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A4: num,B3: num] :
              ( X4
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_865_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X4: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X4
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X4
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X4
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_866_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X4: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X4
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X4
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
              ( X4
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_867_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X4: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X4
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X4
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X3: num,Y3: num] :
              ( X4
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_868_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_869_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_870_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A4: $o,Uw2: $o] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A4: $o,B3: $o,Va: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X4
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X4
                     != ( 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_871_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B3: $o] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N2: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X4
                   != ( 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_872_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,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( 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_873_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,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( 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_874_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_nat,R: vEBT_VEBT > nat > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_875_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_nat,R: real > nat > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_876_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_int,R: vEBT_VEBT > int > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_877_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_int,R: real > int > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_878_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_complex,R: vEBT_VEBT > complex > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A3 )
             => ? [Xa2: complex] :
                  ( ( member_complex @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_879_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_complex,R: real > complex > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A3 )
             => ? [Xa2: complex] :
                  ( ( member_complex @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_880_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_Code_integer,R: vEBT_VEBT > code_integer > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B4 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_881_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_Code_integer,R: real > code_integer > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B4 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_882_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B4: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_883_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B4: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A3 )
                        & ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_884_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_12: list_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_885_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_12: set_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_886_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_12: nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_887_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_12: int] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_888_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_12: complex] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_889_not__finite__existsD,axiom,
    ! [P: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
     => ? [X_12: code_integer] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_890_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X3 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_891_finite__has__minimal2,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A @ A3 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A3 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_892_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_893_finite__has__minimal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ A @ A3 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_894_finite__has__minimal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A @ A3 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_895_finite__has__minimal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_896_finite__has__minimal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_897_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_898_finite__has__minimal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_899_finite__has__maximal2,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A @ A3 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A3 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_900_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_901_finite__has__maximal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ A @ A3 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_902_finite__has__maximal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A @ A3 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_903_finite__has__maximal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_904_finite__has__maximal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_905_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_906_finite__has__maximal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_907_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_908_infinite__imp__nonempty,axiom,
    ! [S3: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S3 )
     => ( S3 != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
thf(fact_909_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ ( finite_finite_real @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_910_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ ( finite_finite_o @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_911_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_912_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_913_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_914_finite_OemptyI,axiom,
    finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).

% finite.emptyI
thf(fact_915_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_916_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_917_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_918_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_919_rev__finite__subset,axiom,
    ! [B4: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A3 @ B4 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_920_rev__finite__subset,axiom,
    ! [B4: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B4 )
       => ( finite3207457112153483333omplex @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_921_rev__finite__subset,axiom,
    ! [B4: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B4 )
       => ( finite6017078050557962740nteger @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_922_rev__finite__subset,axiom,
    ! [B4: set_int,A3: set_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A3 @ B4 )
       => ( finite_finite_int @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_923_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_924_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_925_infinite__super,axiom,
    ! [S3: set_Code_integer,T3: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S3 @ T3 )
     => ( ~ ( finite6017078050557962740nteger @ S3 )
       => ~ ( finite6017078050557962740nteger @ T3 ) ) ) ).

% infinite_super
thf(fact_926_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_927_finite__subset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_928_finite__subset,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ A3 @ B4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( finite3207457112153483333omplex @ A3 ) ) ) ).

% finite_subset
thf(fact_929_finite__subset,axiom,
    ! [A3: set_Code_integer,B4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A3 @ B4 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( finite6017078050557962740nteger @ A3 ) ) ) ).

% finite_subset
thf(fact_930_finite__subset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( finite_finite_int @ B4 )
       => ( finite_finite_int @ A3 ) ) ) ).

% finite_subset
thf(fact_931_finite__has__minimal,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_932_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A3 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_933_finite__has__minimal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_934_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_935_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_936_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_937_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_938_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_939_finite__has__maximal,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_940_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A3 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_941_finite__has__maximal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_942_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_943_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_944_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_945_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_946_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_947_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,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A4: $o,B3: $o] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A4: $o,B3: $o,N2: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X4
                 != ( 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] :
                    ( X4
                   != ( 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] :
                      ( X4
                     != ( 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_948_vebt__maxt_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X4 )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A4
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A4
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_949_vebt__mint_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X4 )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( A4
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A4
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_950_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
      = ( ord_less_eq_set_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_951_pred__subset__eq,axiom,
    ! [R: set_VEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( ord_le418104280809901481VEBT_o
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ R )
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ S3 ) )
      = ( ord_le4337996190870823476T_VEBT @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_952_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S3 ) )
      = ( ord_less_eq_set_real @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_953_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
        @ ^ [X: set_nat] : ( member_set_nat @ X @ S3 ) )
      = ( ord_le6893508408891458716et_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_954_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R )
        @ ^ [X: int] : ( member_int @ X @ S3 ) )
      = ( ord_less_eq_set_int @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_955_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_956_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_957_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_958_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_959_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_960_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_961_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_962_order__refl,axiom,
    ! [X4: set_int] : ( ord_less_eq_set_int @ X4 @ X4 ) ).

% order_refl
thf(fact_963_order__refl,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ X4 @ X4 ) ).

% order_refl
thf(fact_964_order__refl,axiom,
    ! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).

% order_refl
thf(fact_965_order__refl,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_966_order__refl,axiom,
    ! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).

% order_refl
thf(fact_967_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A2: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_968_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A4: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A4 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_969_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A4: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_970_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_971_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_972_false__rule,axiom,
    ! [C: heap_T2636463487746394924on_nat,Q: option_nat > assn] : ( hoare_7629718768684598413on_nat @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_973_false__rule,axiom,
    ! [C: heap_Time_Heap_o,Q: $o > assn] : ( hoare_hoare_triple_o @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_974_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] : ( hoare_1429296392585015714_VEBTi @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_975_false__rule,axiom,
    ! [C: heap_Time_Heap_nat,Q: nat > assn] : ( hoare_3067605981109127869le_nat @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_976_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_977_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_978_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_979_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_980_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_981_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_982_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_983_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_984_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_985_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_986_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_987_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_988_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_989_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_990_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_991_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_992_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_993_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_994_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_995_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_996_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_997_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_998_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_999_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_1000_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1001_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1002_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1003_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_1004_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_1005_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1006_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1007_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_1008_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1009_one__reorient,axiom,
    ! [X4: assn] :
      ( ( one_one_assn = X4 )
      = ( X4 = one_one_assn ) ) ).

% one_reorient
thf(fact_1010_one__reorient,axiom,
    ! [X4: real] :
      ( ( one_one_real = X4 )
      = ( X4 = one_one_real ) ) ).

% one_reorient
thf(fact_1011_one__reorient,axiom,
    ! [X4: rat] :
      ( ( one_one_rat = X4 )
      = ( X4 = one_one_rat ) ) ).

% one_reorient
thf(fact_1012_one__reorient,axiom,
    ! [X4: nat] :
      ( ( one_one_nat = X4 )
      = ( X4 = one_one_nat ) ) ).

% one_reorient
thf(fact_1013_one__reorient,axiom,
    ! [X4: int] :
      ( ( one_one_int = X4 )
      = ( X4 = one_one_int ) ) ).

% one_reorient
thf(fact_1014_bot__empty__eq,axiom,
    ( bot_bot_VEBT_VEBT_o
    = ( ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% bot_empty_eq
thf(fact_1015_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1016_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1017_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_1018_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1019_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1020_bot__set__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex @ bot_bot_complex_o ) ) ).

% bot_set_def
thf(fact_1021_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_1022_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_1023_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_1024_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_1025_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_1026_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_1027_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1028_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
thf(fact_1029_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
thf(fact_1030_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1031_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1032_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1033_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1034_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1035_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1036_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1037_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1038_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1039_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_1040_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1041_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_1042_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_1043_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_1044_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_1045_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_1046_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1047_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1048_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1049_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1050_bot__empty__eq2,axiom,
    ( bot_bo1565574316222977092_nat_o
    = ( ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ bot_bo1642239108664514429BT_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1051_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_1052_bot__empty__eq2,axiom,
    ( bot_bo8134993004553108152eger_o
    = ( ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ bot_bo4276436098303576167nteger ) ) ) ).

% bot_empty_eq2
thf(fact_1053_bot__empty__eq2,axiom,
    ( bot_bot_nat_num_o
    = ( ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ bot_bo7038385379056416535at_num ) ) ) ).

% bot_empty_eq2
thf(fact_1054_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1055_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1056_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1057_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1058_lambda__one,axiom,
    ( ( ^ [X: assn] : X )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_1059_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1069_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1070_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1071_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1072_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1073_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1074_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1075_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1076_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1077_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1078_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1079_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1080_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_1081_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_1082_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_1083_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_1084_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1085_nat__geq__1__eq__neqz,axiom,
    ! [X4: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X4 )
      = ( X4 != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_1086_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_1087_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_1088_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_1089_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_1090_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_1091_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_1092_le__cases3,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X4 @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X4 )
         => ~ ( ord_less_eq_rat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X4 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X4 ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X4 )
                 => ~ ( ord_less_eq_rat @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1093_le__cases3,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X4 @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X4 )
         => ~ ( ord_less_eq_num @ X4 @ Z ) )
       => ( ( ( ord_less_eq_num @ X4 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X4 ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X4 )
                 => ~ ( ord_less_eq_num @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1094_le__cases3,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X4 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X4 )
         => ~ ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X4 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X4 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X4 )
                 => ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1095_le__cases3,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X4 @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X4 )
         => ~ ( ord_less_eq_int @ X4 @ Z ) )
       => ( ( ( ord_less_eq_int @ X4 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X4 ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X4 )
                 => ~ ( ord_less_eq_int @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1096_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [X: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y4 )
          & ( ord_less_eq_set_int @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1097_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [X: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X @ Y4 )
          & ( ord_less_eq_rat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1098_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1099_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1100_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1101_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1102_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1103_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1104_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1105_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1106_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1107_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1108_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1109_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1110_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1111_order__antisym,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1112_order__antisym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1113_order__antisym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1114_order__antisym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1115_order__antisym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1116_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_1117_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_1118_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_1119_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_1120_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_1121_order__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1122_order__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1123_order__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1124_order__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1125_order__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1126_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: rat,B3: rat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1127_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( ( ord_less_eq_num @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: num,B3: num] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1128_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1129_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( ( ord_less_eq_int @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: int,B3: int] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1130_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A2 )
          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1131_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1132_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1133_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1134_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1135_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_1136_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_1137_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_1138_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_1139_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_1140_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1141_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1142_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1143_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1144_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1145_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_1146_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1147_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1148_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1149_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1150_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A2 @ B2 )
          & ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1151_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1152_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1153_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1154_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1155_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1156_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1157_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1158_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1159_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1160_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1161_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1162_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1163_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1164_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1165_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1166_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1167_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1168_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1169_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1170_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1171_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1172_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1173_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1174_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1175_order__eq__refl,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( X4 = Y )
     => ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1176_order__eq__refl,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 = Y )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1177_order__eq__refl,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 = Y )
     => ( ord_less_eq_num @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1178_order__eq__refl,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 = Y )
     => ( ord_less_eq_nat @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1179_order__eq__refl,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 = Y )
     => ( ord_less_eq_int @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1180_linorder__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
      | ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1181_linorder__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
      | ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1182_linorder__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
      | ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1183_linorder__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
      | ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1184_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1185_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1186_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1187_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1188_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1189_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1190_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1191_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1192_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1193_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1194_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1195_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1196_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1197_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1198_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1199_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1200_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1201_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1202_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1203_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1204_linorder__le__cases,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1205_linorder__le__cases,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X4 @ Y )
     => ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1206_linorder__le__cases,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1207_linorder__le__cases,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X4 @ Y )
     => ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1208_order__antisym__conv,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ( ( ord_less_eq_set_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1209_order__antisym__conv,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1210_order__antisym__conv,axiom,
    ! [Y: num,X4: num] :
      ( ( ord_less_eq_num @ Y @ X4 )
     => ( ( ord_less_eq_num @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1211_order__antisym__conv,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ( ( ord_less_eq_nat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1212_order__antisym__conv,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ( ( ord_less_eq_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1213_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1214_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1215_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1216_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1217_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_1218_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_1219_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_1220_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_1221_mult__right__le__one__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1222_mult__right__le__one__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1223_mult__right__le__one__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1224_mult__left__le__one__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1225_mult__left__le__one__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1226_mult__left__le__one__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1227_lt__ex,axiom,
    ! [X4: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1228_lt__ex,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1229_lt__ex,axiom,
    ! [X4: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1230_gt__ex,axiom,
    ! [X4: real] :
    ? [X_12: real] : ( ord_less_real @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1231_gt__ex,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1232_gt__ex,axiom,
    ! [X4: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1233_gt__ex,axiom,
    ! [X4: int] :
    ? [X_12: int] : ( ord_less_int @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1234_dense,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ? [Z3: real] :
          ( ( ord_less_real @ X4 @ Z3 )
          & ( ord_less_real @ Z3 @ Y ) ) ) ).

% dense
thf(fact_1235_dense,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X4 @ Z3 )
          & ( ord_less_rat @ Z3 @ Y ) ) ) ).

% dense
thf(fact_1236_less__imp__neq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1237_less__imp__neq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1238_less__imp__neq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1239_less__imp__neq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1240_less__imp__neq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1241_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_1242_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_1243_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_1244_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1245_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_1246_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1247_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1248_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1249_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1250_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1251_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1252_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1253_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1254_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1255_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1256_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1257_antisym__conv3,axiom,
    ! [Y: real,X4: real] :
      ( ~ ( ord_less_real @ Y @ X4 )
     => ( ( ~ ( ord_less_real @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1258_antisym__conv3,axiom,
    ! [Y: rat,X4: rat] :
      ( ~ ( ord_less_rat @ Y @ X4 )
     => ( ( ~ ( ord_less_rat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1259_antisym__conv3,axiom,
    ! [Y: num,X4: num] :
      ( ~ ( ord_less_num @ Y @ X4 )
     => ( ( ~ ( ord_less_num @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1260_antisym__conv3,axiom,
    ! [Y: nat,X4: nat] :
      ( ~ ( ord_less_nat @ Y @ X4 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1261_antisym__conv3,axiom,
    ! [Y: int,X4: int] :
      ( ~ ( ord_less_int @ Y @ X4 )
     => ( ( ~ ( ord_less_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1262_linorder__cases,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1263_linorder__cases,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1264_linorder__cases,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1265_linorder__cases,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1266_linorder__cases,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1267_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_1268_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1269_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_1270_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1271_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_1272_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_1273_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1274_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_1275_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1276_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_1277_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ? [N4: nat] :
          ( ( P2 @ N4 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ~ ( P2 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1278_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A4: real,B3: real] :
          ( ( ord_less_real @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: real] : ( P @ A4 @ A4 )
       => ( ! [A4: real,B3: real] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1279_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( ( ord_less_rat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: rat] : ( P @ A4 @ A4 )
       => ( ! [A4: rat,B3: rat] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1280_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( ( ord_less_num @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: num] : ( P @ A4 @ A4 )
       => ( ! [A4: num,B3: num] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1281_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1282_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( ( ord_less_int @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B3: int] :
              ( ( P @ B3 @ A4 )
             => ( P @ A4 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1283_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1284_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1285_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1286_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1287_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1288_not__less__iff__gr__or__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y ) )
      = ( ( ord_less_real @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1289_not__less__iff__gr__or__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y ) )
      = ( ( ord_less_rat @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1290_not__less__iff__gr__or__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y ) )
      = ( ( ord_less_num @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1291_not__less__iff__gr__or__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y ) )
      = ( ( ord_less_nat @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1292_not__less__iff__gr__or__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y ) )
      = ( ( ord_less_int @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1293_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1294_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1295_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1296_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1297_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1298_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1299_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1300_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1301_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1302_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1303_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_1304_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_1305_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_1306_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_1307_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_1308_linorder__neqE,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1309_linorder__neqE,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_rat @ X4 @ Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1310_linorder__neqE,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_num @ X4 @ Y )
       => ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1311_linorder__neqE,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_nat @ X4 @ Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1312_linorder__neqE,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_int @ X4 @ Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1313_order__less__asym,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1314_order__less__asym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1315_order__less__asym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1316_order__less__asym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1317_order__less__asym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1318_linorder__neq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
      = ( ( ord_less_real @ X4 @ Y )
        | ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1319_linorder__neq__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
      = ( ( ord_less_rat @ X4 @ Y )
        | ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1320_linorder__neq__iff,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 != Y )
      = ( ( ord_less_num @ X4 @ Y )
        | ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1321_linorder__neq__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
      = ( ( ord_less_nat @ X4 @ Y )
        | ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1322_linorder__neq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
      = ( ( ord_less_int @ X4 @ Y )
        | ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1323_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_1324_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1325_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_1326_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1327_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_1328_order__less__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1329_order__less__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1330_order__less__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1331_order__less__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1332_order__less__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1333_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1334_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1335_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1336_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1337_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1338_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1339_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1340_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1341_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1342_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1343_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1344_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1345_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1346_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1347_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1348_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1349_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1350_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1351_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1352_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1353_order__less__irrefl,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1354_order__less__irrefl,axiom,
    ! [X4: rat] :
      ~ ( ord_less_rat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1355_order__less__irrefl,axiom,
    ! [X4: num] :
      ~ ( ord_less_num @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1356_order__less__irrefl,axiom,
    ! [X4: nat] :
      ~ ( ord_less_nat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1357_order__less__irrefl,axiom,
    ! [X4: int] :
      ~ ( ord_less_int @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1358_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1359_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1360_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1361_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1362_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1363_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1364_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1365_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1366_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1367_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1368_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1369_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1370_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1371_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1372_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1373_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1374_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1375_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1376_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1377_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1378_order__less__not__sym,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1379_order__less__not__sym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1380_order__less__not__sym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1381_order__less__not__sym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1382_order__less__not__sym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1383_order__less__imp__triv,axiom,
    ! [X4: real,Y: real,P: $o] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1384_order__less__imp__triv,axiom,
    ! [X4: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1385_order__less__imp__triv,axiom,
    ! [X4: num,Y: num,P: $o] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1386_order__less__imp__triv,axiom,
    ! [X4: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1387_order__less__imp__triv,axiom,
    ! [X4: int,Y: int,P: $o] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1388_linorder__less__linear,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_real @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1389_linorder__less__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1390_linorder__less__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_num @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1391_linorder__less__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1392_linorder__less__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_int @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1393_order__less__imp__not__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1394_order__less__imp__not__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1395_order__less__imp__not__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1396_order__less__imp__not__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1397_order__less__imp__not__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1398_order__less__imp__not__eq2,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1399_order__less__imp__not__eq2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1400_order__less__imp__not__eq2,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1401_order__less__imp__not__eq2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1402_order__less__imp__not__eq2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1403_order__less__imp__not__less,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1404_order__less__imp__not__less,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1405_order__less__imp__not__less,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1406_order__less__imp__not__less,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1407_order__less__imp__not__less,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1408_subrelI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_1409_subrelI,axiom,
    ! [R3: set_Pr7556676689462069481BT_nat,S2: set_Pr7556676689462069481BT_nat] :
      ( ! [X3: vEBT_VEBT,Y3: nat] :
          ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ R3 )
         => ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le3442269383143156041BT_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_1410_subrelI,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R3 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le2843351958646193337nt_int @ R3 @ S2 ) ) ).

% subrelI
thf(fact_1411_subrelI,axiom,
    ! [R3: set_Pr4811707699266497531nteger,S2: set_Pr4811707699266497531nteger] :
      ( ! [X3: code_integer,Y3: code_integer] :
          ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ R3 )
         => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le3725938330318615451nteger @ R3 @ S2 ) ) ).

% subrelI
thf(fact_1412_subrelI,axiom,
    ! [R3: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Y3: num] :
          ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X3 @ Y3 ) @ R3 )
         => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le8085105155179020875at_num @ R3 @ S2 ) ) ).

% subrelI
thf(fact_1413_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1414_bounded__Max__nat,axiom,
    ! [P: nat > $o,X4: nat,M8: nat] :
      ( ( P @ X4 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M8 ) )
       => ~ ! [M2: nat] :
              ( ( P @ M2 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1415_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = ( ( ( X4 = zero_zero_nat )
         => A )
        & ( ( X4 != zero_zero_nat )
         => ( ( ( X4 = one_one_nat )
             => B )
            & ( X4 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1416_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = ( ( ( X4 = zero_zero_nat )
         => A )
        & ( ( X4 != zero_zero_nat )
         => ( ( ( X4 = one_one_nat )
             => B )
            & ( X4 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1417_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1418_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1419_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1420_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1421_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1422_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1423_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1424_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1425_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1426_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1427_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1428_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1429_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1430_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1431_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1432_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1433_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1434_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1435_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1436_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1437_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1438_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1439_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1440_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1441_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_1442_pred__subset__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ( ord_le1182472622972956176_nat_o
        @ ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ R )
        @ ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ S3 ) )
      = ( ord_le3442269383143156041BT_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_1443_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R )
        @ ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_1444_pred__subset__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ( ord_le3602516367967493612eger_o
        @ ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ R )
        @ ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ S3 ) )
      = ( ord_le3725938330318615451nteger @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_1445_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ord_le3404735783095501756_num_o
        @ ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ S3 ) )
      = ( ord_le8085105155179020875at_num @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_1446_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R ) )
        = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1447_pred__equals__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ( ( ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ R ) )
        = ( ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1448_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R ) )
        = ( ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1449_pred__equals__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ( ( ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ R ) )
        = ( ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1450_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ( ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ R ) )
        = ( ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1451_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_1452_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_1453_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1454_leD,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ Y @ X4 )
     => ~ ( ord_less_real @ X4 @ Y ) ) ).

% leD
thf(fact_1455_leD,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ~ ( ord_less_set_int @ X4 @ Y ) ) ).

% leD
thf(fact_1456_leD,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ~ ( ord_less_rat @ X4 @ Y ) ) ).

% leD
thf(fact_1457_leD,axiom,
    ! [Y: num,X4: num] :
      ( ( ord_less_eq_num @ Y @ X4 )
     => ~ ( ord_less_num @ X4 @ Y ) ) ).

% leD
thf(fact_1458_leD,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ~ ( ord_less_nat @ X4 @ Y ) ) ).

% leD
thf(fact_1459_leD,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ~ ( ord_less_int @ X4 @ Y ) ) ).

% leD
thf(fact_1460_leI,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ord_less_eq_real @ Y @ X4 ) ) ).

% leI
thf(fact_1461_leI,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ Y @ X4 ) ) ).

% leI
thf(fact_1462_leI,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ord_less_eq_num @ Y @ X4 ) ) ).

% leI
thf(fact_1463_leI,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ Y @ X4 ) ) ).

% leI
thf(fact_1464_leI,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ord_less_eq_int @ Y @ X4 ) ) ).

% leI
thf(fact_1465_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1466_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_1467_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1468_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1469_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1470_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1471_antisym__conv1,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_eq_real @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1472_antisym__conv1,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1473_antisym__conv1,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1474_antisym__conv1,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1475_antisym__conv1,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1476_antisym__conv1,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_1477_antisym__conv2,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ~ ( ord_less_real @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1478_antisym__conv2,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ~ ( ord_less_set_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1479_antisym__conv2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ~ ( ord_less_rat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1480_antisym__conv2,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ~ ( ord_less_num @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1481_antisym__conv2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ~ ( ord_less_nat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1482_antisym__conv2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ~ ( ord_less_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_1483_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_1484_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_1485_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_1486_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_1487_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_eq_real @ X @ Y4 )
          & ~ ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1488_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y4 )
          & ~ ( ord_less_eq_set_int @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1489_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X @ Y4 )
          & ~ ( ord_less_eq_rat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1490_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ~ ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1491_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1492_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1493_not__le__imp__less,axiom,
    ! [Y: real,X4: real] :
      ( ~ ( ord_less_eq_real @ Y @ X4 )
     => ( ord_less_real @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_1494_not__le__imp__less,axiom,
    ! [Y: rat,X4: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X4 )
     => ( ord_less_rat @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_1495_not__le__imp__less,axiom,
    ! [Y: num,X4: num] :
      ( ~ ( ord_less_eq_num @ Y @ X4 )
     => ( ord_less_num @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_1496_not__le__imp__less,axiom,
    ! [Y: nat,X4: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X4 )
     => ( ord_less_nat @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_1497_not__le__imp__less,axiom,
    ! [Y: int,X4: int] :
      ( ~ ( ord_less_eq_int @ Y @ X4 )
     => ( ord_less_int @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_1498_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_real @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1499_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( ord_less_set_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1500_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_rat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1501_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_num @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1502_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1503_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1504_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1505_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1506_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1507_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1508_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1509_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1510_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1511_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1512_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1513_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1514_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1515_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1516_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1517_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1518_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1519_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1520_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1521_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1522_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ A2 @ B2 )
          & ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1523_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A2 @ B2 )
          & ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1524_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1525_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1526_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1527_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1528_dense__ge__bounded,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ Z @ X4 )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z @ W )
           => ( ( ord_less_real @ W @ X4 )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1529_dense__ge__bounded,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X4 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z @ W )
           => ( ( ord_less_rat @ W @ X4 )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1530_dense__le__bounded,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X4 @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1531_dense__le__bounded,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X4 @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1532_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_real @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1533_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B2: set_int,A2: set_int] :
          ( ( ord_less_set_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1534_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_rat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1535_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_num @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1536_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1537_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1538_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_eq_real @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1539_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A2: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1540_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1541_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1542_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1543_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1544_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1545_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1546_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1547_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1548_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1549_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1550_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1551_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1552_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1553_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1554_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1555_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1556_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_eq_real @ B2 @ A2 )
          & ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1557_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A2: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A2 )
          & ~ ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1558_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1559_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1560_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1561_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1562_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_1563_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_1564_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_1565_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_1566_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_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_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_1574_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_real @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1575_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y4: set_int] :
          ( ( ord_less_set_int @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1576_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y4: rat] :
          ( ( ord_less_rat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1577_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_num @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1578_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_nat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1579_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_int @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1580_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_eq_real @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1581_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1582_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1583_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y4: num] :
          ( ( ord_less_eq_num @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1584_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1585_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y4: int] :
          ( ( ord_less_eq_int @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1586_linorder__not__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X4 @ Y ) )
      = ( ord_less_real @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1587_linorder__not__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X4 @ Y ) )
      = ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1588_linorder__not__le,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X4 @ Y ) )
      = ( ord_less_num @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1589_linorder__not__le,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
      = ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1590_linorder__not__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X4 @ Y ) )
      = ( ord_less_int @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1591_linorder__not__less,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y ) )
      = ( ord_less_eq_real @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1592_linorder__not__less,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y ) )
      = ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1593_linorder__not__less,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y ) )
      = ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1594_linorder__not__less,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1595_linorder__not__less,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y ) )
      = ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1596_order__less__imp__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1597_order__less__imp__le,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X4 @ Y )
     => ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1598_order__less__imp__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1599_order__less__imp__le,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ord_less_eq_num @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1600_order__less__imp__le,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1601_order__less__imp__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ord_less_eq_int @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_order__le__less__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1615_order__le__less__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1616_order__le__less__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1617_order__le__less__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1618_order__le__less__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1619_order__le__less__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1620_order__less__le__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1621_order__less__le__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1622_order__less__le__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1623_order__less__le__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1624_order__less__le__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1625_order__less__le__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1626_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1627_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1628_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1629_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1630_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1631_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1632_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1633_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1634_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1635_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1636_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1637_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1638_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1639_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1640_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1641_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1642_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1643_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1644_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1645_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1646_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1647_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1648_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1649_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1650_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1651_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1652_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1653_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1654_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1655_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1656_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1657_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1658_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1659_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1660_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1661_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1662_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1663_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1664_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1665_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1666_linorder__le__less__linear,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
      | ( ord_less_real @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1667_linorder__le__less__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
      | ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1668_linorder__le__less__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
      | ( ord_less_num @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1669_linorder__le__less__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
      | ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1670_linorder__le__less__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
      | ( ord_less_int @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1671_order__le__imp__less__or__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ord_less_real @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1672_order__le__imp__less__or__eq,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_set_int @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1673_order__le__imp__less__or__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_rat @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1674_order__le__imp__less__or__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_num @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1675_order__le__imp__less__or__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_nat @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1676_order__le__imp__less__or__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_int @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1677_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_1678_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_1679_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1680_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1681_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1682_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_1683_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_1684_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_1685_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_1686_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1687_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_1688_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_1689_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_1690_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_1691_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1692_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1693_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1694_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1695_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1696_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1697_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1698_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_1699_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_1700_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_1701_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1702_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_1703_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N7 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1704_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N8 )
         => ( ord_less_nat @ X @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1705_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N8 )
         => ( ord_less_eq_nat @ X @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1706_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R2: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_1707_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R2: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_1708_field__le__mult__one__interval,axiom,
    ! [X4: 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 @ X4 ) @ Y ) ) )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_1709_field__le__mult__one__interval,axiom,
    ! [X4: 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 @ X4 ) @ Y ) ) )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_1710_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X4: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1711_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X4: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1712_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X4: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1713_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X4: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1714_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X4: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1715_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X4: code_integer > real,Y: code_integer > real] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( times_times_real @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1716_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X4: vEBT_VEBT > rat,Y: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1717_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X4: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1718_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X4: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1719_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X4: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_1720_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1721_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1722_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1723_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1724_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_1725_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,X4: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X4 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1726_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,X4: 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 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_1727_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,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X4 )
        = Y )
     => ( ( ( X4
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv2: $o] :
                ( X4
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] :
                  ( X4
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X4
                      = ( 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_1728_option_Osize__gen_I1_J,axiom,
    ! [X4: nat > nat] :
      ( ( size_option_nat @ X4 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1729_option_Osize__gen_I1_J,axiom,
    ! [X4: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X4 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1730_option_Osize__gen_I1_J,axiom,
    ! [X4: num > nat] :
      ( ( size_option_num @ X4 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1731_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_1732_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_1733_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_1734_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_1735_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1736_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1737_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1738_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1739_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% norm_assertion_simps(2)
thf(fact_1740_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% norm_assertion_simps(1)
thf(fact_1741_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_1742_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_1743_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_1744_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_1745_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_1746_linordered__field__no__ub,axiom,
    ! [X6: real] :
    ? [X_12: real] : ( ord_less_real @ X6 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1747_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X6 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1748_linordered__field__no__lb,axiom,
    ! [X6: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X6 ) ).

% linordered_field_no_lb
thf(fact_1749_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X6 ) ).

% linordered_field_no_lb
thf(fact_1750_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_1751_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_1752_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_1753_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_1754_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,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_1755_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_1756_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_1757_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_1758_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_1759_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_1760_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_1761_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_1762_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_1763_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_1764_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_1765_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_1766_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_1767_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_1768_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_1769_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_1770_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1771_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1772_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1773_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1774_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1775_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1776_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1777_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1778_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_1779_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1780_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1781_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1782_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_1783_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1784_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1785_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1786_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_1787_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,X4: 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 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_1788_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,X4: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X4 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1789_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_1790_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_1791_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_1792_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_1793_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_o @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_1794_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_1795_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_1796_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_1797_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_o @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_1798_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_1799_Collect__empty__eq__bot,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( P = bot_bot_complex_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1800_Collect__empty__eq__bot,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( P = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1801_Collect__empty__eq__bot,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( P = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1802_Collect__empty__eq__bot,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( P = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1803_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1804_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1805_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1806_htt__htD,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ P @ C @ Q @ T )
     => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_1807_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ P @ C @ Q @ T )
     => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_1808_htt__htD,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T )
     => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_1809_htt__htD,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ P @ C @ Q @ T )
     => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ).

% htt_htD
thf(fact_1810_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_1811_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_1812_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_1813_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_1814_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_1815_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_1816_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_1817_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_1818_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_1819_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_1820_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_1821_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_1822_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_1823_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_1824_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_1825_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_1826_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1827_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_1828_vebt__buildupi__refines,axiom,
    ! [N: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_V739175172307565963ildupi @ N ) ) ).

% vebt_buildupi_refines
thf(fact_1829_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% TBOUND_vebt_buildupi
thf(fact_1830_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N4: nat] :
            ( ( ord_less_eq_nat @ M5 @ N4 )
            & ( member_nat @ N4 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1831_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( ( ord_less_nat @ K @ M2 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M2 @ N9 )
              & ( member_nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_1832_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N4: nat] :
            ( ( ord_less_nat @ M5 @ N4 )
            & ( member_nat @ N4 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1833_VEBT__internal_Oheight_Ocases,axiom,
    ! [X4: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] :
          ( X4
         != ( vEBT_Leaf @ A4 @ B3 ) )
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X4
           != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_1834_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_1835_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_1836_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X6: complex] :
              ( ( member_complex @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1837_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic2659822949269061924r_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1838_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X6: real] :
              ( ( member_real @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1839_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X6: $o] :
              ( ( member_o @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1840_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1841_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X6: int] :
              ( ( member_int @ X6 @ S3 )
              & ( ord_less_real @ ( F @ X6 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1842_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X6: complex] :
              ( ( member_complex @ X6 @ S3 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1843_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ~ ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ S3 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( F @ ( lattic5439806495466278992er_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1844_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X6: real] :
              ( ( member_real @ X6 @ S3 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1845_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X6: $o] :
              ( ( member_o @ X6 @ S3 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1846_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F12: $o > $o > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A4: $o,B3: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A4 @ B3 ) @ ( F12 @ A4 @ B3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_1847_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_Time_Heap_o,F12: $o > $o > heap_Time_Heap_o,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( Ti = Ti2 )
     => ( ! [A4: $o,B3: $o] : ( refine_Imp_refines_o @ ( F1 @ A4 @ B3 ) @ ( F12 @ A4 @ B3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine_Imp_refines_o @ ( vEBT_c6104975476656191286Heap_o @ F22 @ F1 @ Ti ) @ ( vEBT_c6104975476656191286Heap_o @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_1848_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T5737551269749752165_VEBTi @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_1849_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T8353473612707095248on_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_1850_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_o @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_o @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_1851_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_1852_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H3 ) @ T )
     => ( time_T5737551269749752165_VEBTi @ M @ T ) ) ).

% TBOUNDI
thf(fact_1853_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H3 ) @ T )
     => ( time_T8353473612707095248on_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_1854_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M @ H3 ) @ T )
     => ( time_TBOUND_o @ M @ T ) ) ).

% TBOUNDI
thf(fact_1855_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M @ H3 ) @ T )
     => ( time_TBOUND_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_1856_TBOUND__def,axiom,
    ( time_T5737551269749752165_VEBTi
    = ( ^ [M5: heap_T8145700208782473153_VEBTi,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_1857_TBOUND__def,axiom,
    ( time_T8353473612707095248on_nat
    = ( ^ [M5: heap_T2636463487746394924on_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_1858_TBOUND__def,axiom,
    ( time_TBOUND_o
    = ( ^ [M5: heap_Time_Heap_o,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_1859_TBOUND__def,axiom,
    ( time_TBOUND_nat
    = ( ^ [M5: heap_Time_Heap_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_1860_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T: nat,T4: nat] :
      ( ( time_T5737551269749752165_VEBTi @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T4 )
       => ( time_T5737551269749752165_VEBTi @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_1861_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,T: nat,T4: nat] :
      ( ( time_T8353473612707095248on_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T4 )
       => ( time_T8353473612707095248on_nat @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_1862_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,T: nat,T4: nat] :
      ( ( time_TBOUND_o @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T4 )
       => ( time_TBOUND_o @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_1863_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,T: nat,T4: nat] :
      ( ( time_TBOUND_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T4 )
       => ( time_TBOUND_nat @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_1864_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( time_T8353473612707095248on_nat @ C @ T )
       => ( time_htt_option_nat @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_1865_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( time_TBOUND_o @ C @ T )
       => ( time_htt_o @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_1866_httI__TBOUND,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( time_T5737551269749752165_VEBTi @ C @ T )
       => ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_1867_httI__TBOUND,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( time_TBOUND_nat @ C @ T )
       => ( time_htt_nat @ P @ C @ Q @ T ) ) ) ).

% httI_TBOUND
thf(fact_1868_finite__transitivity__chain,axiom,
    ! [A3: set_VEBT_VEBT,R: vEBT_VEBT > vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ! [X3: vEBT_VEBT] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Z3: vEBT_VEBT] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ A3 )
               => ? [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1869_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ! [X3: set_nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: set_nat] :
                ( ( member_set_nat @ X3 @ A3 )
               => ? [Y5: set_nat] :
                    ( ( member_set_nat @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1870_finite__transitivity__chain,axiom,
    ! [A3: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X3: complex] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: complex,Y3: complex,Z3: complex] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
               => ? [Y5: complex] :
                    ( ( member_complex @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1871_finite__transitivity__chain,axiom,
    ! [A3: set_Code_integer,R: code_integer > code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: code_integer,Y3: code_integer,Z3: code_integer] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ A3 )
               => ? [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1872_finite__transitivity__chain,axiom,
    ! [A3: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X3: real] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: real,Y3: real,Z3: real] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ A3 )
               => ? [Y5: real] :
                    ( ( member_real @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1873_finite__transitivity__chain,axiom,
    ! [A3: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X3: $o] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: $o,Y3: $o,Z3: $o] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ? [Y5: $o] :
                    ( ( member_o @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1874_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1875_finite__transitivity__chain,axiom,
    ! [A3: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: int,Y3: int,Z3: int] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ A3 )
               => ? [Y5: int] :
                    ( ( member_int @ Y5 @ A3 )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1876_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_1877_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_1878_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_1879_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_1880_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_1881_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_1882_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_1883_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_1884_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_1885_arg__min__least,axiom,
    ! [S3: set_Code_integer,Y: code_integer,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( member_Code_integer @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic2632268341708553362er_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1886_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F3: heap_T8145700208782473153_VEBTi,N: nat] :
      ( ( refine5565527176597971370_VEBTi @ F @ F3 )
     => ( refine3700189196150522554_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ F ) @ ( vEBT_V1859673955506687831_VEBTi @ N @ F3 ) ) ) ).

% refines_replicate
thf(fact_1887_refines__replicate,axiom,
    ! [F: heap_Time_Heap_o,F3: heap_Time_Heap_o,N: nat] :
      ( ( refine_Imp_refines_o @ F @ F3 )
     => ( refine5896690332125372649list_o @ ( vEBT_V2326993469660664182atei_o @ N @ F ) @ ( vEBT_V2326993469660664182atei_o @ N @ F3 ) ) ) ).

% refines_replicate
thf(fact_1888_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,C3: heap_T2636463487746394924on_nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( refine7594492741263601813on_nat @ C3 @ C )
       => ( hoare_7629718768684598413on_nat @ P @ C3 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_1889_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,C3: heap_Time_Heap_o] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( refine_Imp_refines_o @ C3 @ C )
       => ( hoare_hoare_triple_o @ P @ C3 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_1890_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,C3: heap_T8145700208782473153_VEBTi] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( refine5565527176597971370_VEBTi @ C3 @ C )
       => ( hoare_1429296392585015714_VEBTi @ P @ C3 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_1891_hoare__triple__refines,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,C3: heap_Time_Heap_nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( refine1365783493865988805es_nat @ C3 @ C )
       => ( hoare_3067605981109127869le_nat @ P @ C3 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_1892_vebt__maxt_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A4
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A4
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_1893_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_1894_vebt__mint_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( A4
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A4
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_1895_subset__emptyI,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ~ ( member_VEBT_VEBT @ X3 @ A3 )
     => ( ord_le4337996190870823476T_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ).

% subset_emptyI
thf(fact_1896_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member_set_nat @ X3 @ A3 )
     => ( ord_le6893508408891458716et_nat @ A3 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1897_subset__emptyI,axiom,
    ! [A3: set_real] :
      ( ! [X3: real] :
          ~ ( member_real @ X3 @ A3 )
     => ( ord_less_eq_set_real @ A3 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_1898_subset__emptyI,axiom,
    ! [A3: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A3 )
     => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_1899_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A3 )
     => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1900_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A3 )
     => ( ord_less_eq_set_int @ A3 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1901_complete__interval,axiom,
    ! [A: real,B: real,P: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C4: real] :
              ( ( ord_less_eq_real @ A @ C4 )
              & ( ord_less_eq_real @ C4 @ B )
              & ! [X6: real] :
                  ( ( ( ord_less_eq_real @ A @ X6 )
                    & ( ord_less_real @ X6 @ C4 ) )
                 => ( P @ X6 ) )
              & ! [D3: real] :
                  ( ! [X3: real] :
                      ( ( ( ord_less_eq_real @ A @ X3 )
                        & ( ord_less_real @ X3 @ D3 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_real @ D3 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1902_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C4: nat] :
              ( ( ord_less_eq_nat @ A @ C4 )
              & ( ord_less_eq_nat @ C4 @ B )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C4 ) )
                 => ( P @ X6 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1903_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C4: int] :
              ( ( ord_less_eq_int @ A @ C4 )
              & ( ord_less_eq_int @ C4 @ B )
              & ! [X6: int] :
                  ( ( ( ord_less_eq_int @ A @ X6 )
                    & ( ord_less_int @ X6 @ C4 ) )
                 => ( P @ X6 ) )
              & ! [D3: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D3 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_int @ D3 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1904_verit__comp__simplify1_I3_J,axiom,
    ! [B6: real,A6: real] :
      ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
      = ( ord_less_real @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1905_verit__comp__simplify1_I3_J,axiom,
    ! [B6: rat,A6: rat] :
      ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
      = ( ord_less_rat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1906_verit__comp__simplify1_I3_J,axiom,
    ! [B6: num,A6: num] :
      ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
      = ( ord_less_num @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1907_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1908_verit__comp__simplify1_I3_J,axiom,
    ! [B6: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
      = ( ord_less_int @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1909_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_eq_real @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1910_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1911_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_eq_num @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1912_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1913_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1914_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_eq_real @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1915_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_eq_rat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1916_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_eq_num @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1917_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1918_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_eq_int @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1919_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X4: nat,T: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X4 ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X4 ) ) ).

% vebt_memberi_refines
thf(fact_1920_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_1921_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1922_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1923_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1924_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1925_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1926_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_1927_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_1928_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_1929_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_1930_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_real @ T @ X6 ) ) ).

% minf(7)
thf(fact_1931_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_rat @ T @ X6 ) ) ).

% minf(7)
thf(fact_1932_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_num @ T @ X6 ) ) ).

% minf(7)
thf(fact_1933_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_1934_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_1935_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_real @ X6 @ T ) ) ).

% minf(5)
thf(fact_1936_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_rat @ X6 @ T ) ) ).

% minf(5)
thf(fact_1937_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_num @ X6 @ T ) ) ).

% minf(5)
thf(fact_1938_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_1939_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_1940_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1941_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1942_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1943_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1944_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1945_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1946_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1947_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1948_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1949_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1950_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1951_minf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1952_minf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1953_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1954_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1955_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1956_minf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1957_minf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1958_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1959_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1960_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_real @ T @ X6 ) ) ).

% pinf(7)
thf(fact_1961_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_rat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_1962_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_num @ T @ X6 ) ) ).

% pinf(7)
thf(fact_1963_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_1964_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_int @ T @ X6 ) ) ).

% pinf(7)
thf(fact_1965_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_real @ X6 @ T ) ) ).

% pinf(5)
thf(fact_1966_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_rat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_1967_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_num @ X6 @ T ) ) ).

% pinf(5)
thf(fact_1968_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_1969_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_int @ X6 @ T ) ) ).

% pinf(5)
thf(fact_1970_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_1971_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_1972_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_1973_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_1974_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_1975_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_1976_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_1977_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_1978_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_1979_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_1980_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1981_pinf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1982_pinf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1983_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1984_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1985_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1986_pinf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1987_pinf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1988_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1989_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P4 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q3 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1990_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1991_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1992_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1993_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1994_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1995_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_1996_prop__restrict,axiom,
    ! [X4: vEBT_VEBT,Z6: set_VEBT_VEBT,X7: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ X4 @ Z6 )
     => ( ( ord_le4337996190870823476T_VEBT @ Z6
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_1997_prop__restrict,axiom,
    ! [X4: real,Z6: set_real,X7: set_real,P: real > $o] :
      ( ( member_real @ X4 @ Z6 )
     => ( ( ord_less_eq_set_real @ Z6
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_1998_prop__restrict,axiom,
    ! [X4: complex,Z6: set_complex,X7: set_complex,P: complex > $o] :
      ( ( member_complex @ X4 @ Z6 )
     => ( ( ord_le211207098394363844omplex @ Z6
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_1999_prop__restrict,axiom,
    ! [X4: list_nat,Z6: set_list_nat,X7: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X4 @ Z6 )
     => ( ( ord_le6045566169113846134st_nat @ Z6
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_2000_prop__restrict,axiom,
    ! [X4: set_nat,Z6: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X4 @ Z6 )
     => ( ( ord_le6893508408891458716et_nat @ Z6
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_2001_prop__restrict,axiom,
    ! [X4: nat,Z6: set_nat,X7: set_nat,P: nat > $o] :
      ( ( member_nat @ X4 @ Z6 )
     => ( ( ord_less_eq_set_nat @ Z6
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_2002_prop__restrict,axiom,
    ! [X4: int,Z6: set_int,X7: set_int,P: int > $o] :
      ( ( member_int @ X4 @ Z6 )
     => ( ( ord_less_eq_set_int @ Z6
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X4 ) ) ) ).

% prop_restrict
thf(fact_2003_Collect__restrict,axiom,
    ! [X7: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2004_Collect__restrict,axiom,
    ! [X7: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2005_Collect__restrict,axiom,
    ! [X7: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2006_Collect__restrict,axiom,
    ! [X7: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2007_Collect__restrict,axiom,
    ! [X7: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2008_Collect__restrict,axiom,
    ! [X7: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2009_Collect__restrict,axiom,
    ! [X7: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_2010_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X6 ) ) ).

% minf(8)
thf(fact_2011_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).

% minf(8)
thf(fact_2012_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X6 ) ) ).

% minf(8)
thf(fact_2013_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_2014_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X6 ) ) ).

% minf(8)
thf(fact_2015_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_eq_real @ X6 @ T ) ) ).

% minf(6)
thf(fact_2016_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_eq_rat @ X6 @ T ) ) ).

% minf(6)
thf(fact_2017_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_eq_num @ X6 @ T ) ) ).

% minf(6)
thf(fact_2018_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_2019_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_eq_int @ X6 @ T ) ) ).

% minf(6)
thf(fact_2020_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,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X4 )
       => ( ( ( X4
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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_2021_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_2022_time__replicate,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,C: nat,N: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X4 @ H3 ) @ C )
     => ( ord_less_eq_nat @ ( time_t3534373299052942712_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% time_replicate
thf(fact_2023_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_2024_TBOUND__replicate,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,C: nat,N: nat] :
      ( ( time_T5737551269749752165_VEBTi @ X4 @ C )
     => ( time_T8149879359713347829_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_2025_TBOUND__replicate,axiom,
    ! [X4: heap_T2636463487746394924on_nat,C: nat,N: nat] :
      ( ( time_T8353473612707095248on_nat @ X4 @ C )
     => ( time_T3808005469503390304on_nat @ ( vEBT_V792416675989592002on_nat @ N @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_2026_TBOUND__replicate,axiom,
    ! [X4: heap_Time_Heap_o,C: nat,N: nat] :
      ( ( time_TBOUND_o @ X4 @ C )
     => ( time_TBOUND_list_o @ ( vEBT_V2326993469660664182atei_o @ N @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_2027_TBOUND__replicate,axiom,
    ! [X4: heap_Time_Heap_nat,C: nat,N: nat] :
      ( ( time_TBOUND_nat @ X4 @ C )
     => ( time_TBOUND_list_nat @ ( vEBT_V7726092123322077554ei_nat @ N @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_2028_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_2029_accp__subset,axiom,
    ! [R1: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,R22: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o] :
      ( ( ord_le1077754993875142464_nat_o @ R1 @ R22 )
     => ( ord_le7812727212727832188_nat_o @ ( accp_P2887432264394892906BT_nat @ R22 ) @ ( accp_P2887432264394892906BT_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2030_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2031_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_2032_accp__subset,axiom,
    ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
      ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
     => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2033_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2034_accp__subset__induct,axiom,
    ! [D4: produc9072475918466114483BT_nat > $o,R: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o,X4: produc9072475918466114483BT_nat,P: produc9072475918466114483BT_nat > $o] :
      ( ( ord_le7812727212727832188_nat_o @ D4 @ ( accp_P2887432264394892906BT_nat @ R ) )
     => ( ! [X3: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( ( D4 @ X3 )
           => ( ( R @ Z3 @ X3 )
             => ( D4 @ Z3 ) ) )
       => ( ( D4 @ X4 )
         => ( ! [X3: produc9072475918466114483BT_nat] :
                ( ( D4 @ X3 )
               => ( ! [Z5: produc9072475918466114483BT_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X4 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2035_accp__subset__induct,axiom,
    ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
     => ( ! [X3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( ( D4 @ X3 )
           => ( ( R @ Z3 @ X3 )
             => ( D4 @ Z3 ) ) )
       => ( ( D4 @ X4 )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( D4 @ X3 )
               => ( ! [Z5: product_prod_nat_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X4 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2036_accp__subset__induct,axiom,
    ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X4: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
     => ( ! [X3: product_prod_int_int,Z3: product_prod_int_int] :
            ( ( D4 @ X3 )
           => ( ( R @ Z3 @ X3 )
             => ( D4 @ Z3 ) ) )
       => ( ( D4 @ X4 )
         => ( ! [X3: product_prod_int_int] :
                ( ( D4 @ X3 )
               => ( ! [Z5: product_prod_int_int] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X4 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2037_accp__subset__induct,axiom,
    ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X4: list_nat,P: list_nat > $o] :
      ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
     => ( ! [X3: list_nat,Z3: list_nat] :
            ( ( D4 @ X3 )
           => ( ( R @ Z3 @ X3 )
             => ( D4 @ Z3 ) ) )
       => ( ( D4 @ X4 )
         => ( ! [X3: list_nat] :
                ( ( D4 @ X3 )
               => ( ! [Z5: list_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X4 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2038_accp__subset__induct,axiom,
    ! [D4: nat > $o,R: nat > nat > $o,X4: nat,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
     => ( ! [X3: nat,Z3: nat] :
            ( ( D4 @ X3 )
           => ( ( R @ Z3 @ X3 )
             => ( D4 @ Z3 ) ) )
       => ( ( D4 @ X4 )
         => ( ! [X3: nat] :
                ( ( D4 @ X3 )
               => ( ! [Z5: nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X4 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2039_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_2040_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_2041_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_2042_even__odd__cases,axiom,
    ! [X4: nat] :
      ( ! [N2: nat] :
          ( X4
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X4
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_2043_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X4 ) ) ).

% min_Null_member
thf(fact_2044_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_2045_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2046_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2047_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2048_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2049_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2050_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2051_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2052_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2053_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R2: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_VEBT_minNull @ T ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_2054_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_2055_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_2056_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2057_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2058_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2059_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_2060_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2061_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2062_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_2063_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_2064_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_2065_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_2066_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_2067_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_2068_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_2069_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_2070_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_2071_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_2072_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ B @ A ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_2073_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_2074_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_2075_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_2076_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_2077_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_2078_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_2079_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_2080_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_2081_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_2082_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y )
        = zero_zero_nat )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_2083_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X4 @ Y ) )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_2084_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_2085_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_2086_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2087_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2088_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2089_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2090_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2091_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2092_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2093_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2094_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2095_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2096_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2097_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2098_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2099_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2100_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2101_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2102_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2103_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2104_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2105_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_2106_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_2107_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2108_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_2109_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R2: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_VEBT_minNull @ T ) ) ) ) ) ).

% minNulli_rule
thf(fact_2110_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_2111_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_2112_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_2113_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_2114_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_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_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_2138_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_2139_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_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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_2147_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_2148_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_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_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_2156_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_2157_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X4 ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_2158_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_2159_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_2160_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_2161_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_2162_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A3 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2163_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2164_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2165_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2166_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_2167_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_2168_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_2169_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_2170_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_2171_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_2172_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_2173_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_2174_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_2175_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_2176_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_2177_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_2178_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_2179_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_2180_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2181_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2182_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2183_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2184_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2185_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2186_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2187_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2188_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2189_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2190_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2191_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2192_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2193_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2194_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2195_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2196_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2197_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_2198_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_2199_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_2200_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_2201_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_2202_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2203_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2204_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2205_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2206_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2207_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2208_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2209_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2210_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_2211_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2212_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2213_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2214_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2228_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2229_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2230_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_2231_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2232_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2233_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2234_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2235_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C4: nat] :
            ( B
           != ( plus_plus_nat @ A @ C4 ) ) ) ).

% less_eqE
thf(fact_2236_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2237_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2238_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2239_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2240_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C5: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_2241_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2242_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2243_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2244_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2245_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2246_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2247_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2248_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2249_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_2250_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_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_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_2260_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_2261_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2262_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2263_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2264_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2265_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2266_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2267_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2268_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2269_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2270_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2271_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2272_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2273_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2274_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2275_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2276_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2277_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2278_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2279_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2280_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2281_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2282_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2283_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2284_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2285_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2286_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2287_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2288_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2289_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2290_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2291_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2292_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2293_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2294_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2295_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2296_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2297_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2298_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% distrib_left
thf(fact_2299_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_2300_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2301_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2302_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_2303_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% distrib_right
thf(fact_2304_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2305_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2306_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2307_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2308_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2309_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_2310_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_2311_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_2312_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_2313_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_2314_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_2315_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_2316_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_2317_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_2318_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_2319_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_2320_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_2321_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_2322_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_2323_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_2324_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_2333_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_2334_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_2335_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_2336_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_2337_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_2338_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_2339_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2340_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2341_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2342_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2343_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2344_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2345_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2346_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2347_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2348_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2349_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2350_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2351_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2352_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2353_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2354_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_add__nonneg__eq__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X4 @ Y )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2364_add__nonneg__eq__0__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X4 @ Y )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2365_add__nonneg__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2366_add__nonneg__eq__0__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X4 @ Y )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2367_add__nonpos__eq__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X4 @ Y )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2368_add__nonpos__eq__0__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X4 @ Y )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2369_add__nonpos__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2370_add__nonpos__eq__0__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X4 @ Y )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2371_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_2372_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_2373_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_2374_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_2375_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_2376_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_2377_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_2378_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_2379_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C4: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C4 ) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2380_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2381_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2382_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2383_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2384_add__less__zeroD,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2385_add__less__zeroD,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X4 @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X4 @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2386_add__less__zeroD,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X4 @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2396_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2397_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2398_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2399_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2400_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2401_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2402_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2403_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2404_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2405_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2406_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_2415_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_2416_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_2417_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).

% less_natE
thf(fact_2418_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_2419_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2420_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2421_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2422_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
         => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2423_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_2424_mlex__snd__decrI,axiom,
    ! [A: nat,A6: nat,B: nat,B6: nat,N7: nat] :
      ( ( A = A6 )
     => ( ( ord_less_nat @ B @ B6 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A6 @ N7 ) @ B6 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_2425_mlex__fst__decrI,axiom,
    ! [A: nat,A6: nat,B: nat,N7: nat,B6: nat] :
      ( ( ord_less_nat @ A @ A6 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ( ord_less_nat @ B6 @ N7 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A6 @ N7 ) @ B6 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_2426_mlex__bound,axiom,
    ! [A: nat,A3: nat,B: nat,N7: nat] :
      ( ( ord_less_nat @ A @ A3 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( times_times_nat @ A3 @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_2427_mlex__leI,axiom,
    ! [A: nat,A6: nat,B: nat,B6: nat,N7: nat] :
      ( ( ord_less_eq_nat @ A @ A6 )
     => ( ( ord_less_eq_nat @ B @ B6 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A6 @ N7 ) @ B6 ) ) ) ) ).

% mlex_leI
thf(fact_2428_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_2429_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_2430_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X4: vEBT_VEBT > complex,Y: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2431_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X4: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2432_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X4: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2433_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X4: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2434_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X4: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2435_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X4: code_integer > complex,Y: code_integer > complex] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X4 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2436_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X4: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2437_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X4: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2438_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X4: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2439_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X4: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X4 @ 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 @ ( X4 @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_2440_field__le__epsilon,axiom,
    ! [X4: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% field_le_epsilon
thf(fact_2441_field__le__epsilon,axiom,
    ! [X4: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% field_le_epsilon
thf(fact_2442_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2443_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2444_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2445_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2446_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2447_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2448_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2449_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2450_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_2451_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_2452_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_2453_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_2454_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_2455_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_2456_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_2457_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_2458_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_2459_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_2460_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_2461_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_2462_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_2463_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_2464_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_2465_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_2466_sum__squares__ge__zero,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2467_sum__squares__ge__zero,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2468_sum__squares__ge__zero,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2469_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2470_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2471_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2472_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2473_not__sum__squares__lt__zero,axiom,
    ! [X4: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2474_not__sum__squares__lt__zero,axiom,
    ! [X4: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2475_not__sum__squares__lt__zero,axiom,
    ! [X4: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2476_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X4 )
     => ( ! [Uv2: $o] :
            ( X4
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X4
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X4
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2477_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X4 )
     => ( ( X4
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2478_convex__bound__le,axiom,
    ! [X4: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X4 @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2479_convex__bound__le,axiom,
    ! [X4: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X4 @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2480_convex__bound__le,axiom,
    ! [X4: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X4 @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2481_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X4 )
        = Y )
     => ( ( ( X4
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X4
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X4
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2482_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_2483_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_2484_convex__bound__lt,axiom,
    ! [X4: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X4 @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2485_convex__bound__lt,axiom,
    ! [X4: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X4 @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2486_convex__bound__lt,axiom,
    ! [X4: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X4 @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2487_option_Osize__gen_I2_J,axiom,
    ! [X4: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X4 @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X4 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2488_option_Osize__gen_I2_J,axiom,
    ! [X4: nat > nat,X2: nat] :
      ( ( size_option_nat @ X4 @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X4 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2489_option_Osize__gen_I2_J,axiom,
    ! [X4: num > nat,X2: num] :
      ( ( size_option_num @ X4 @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X4 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2490_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_2491_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_2492_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_2493_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_2494_sum__squares__eq__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2495_sum__squares__eq__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2496_sum__squares__eq__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2497_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_2498_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_2499_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_2500_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X4: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X4 @ Xa @ Xb )
        = Y )
     => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X4 @ ( produc488173922507101015at_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X4 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ V2 ) )
               => ( ( Xb = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X4 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
           => ~ ! [A4: product_prod_nat_nat] :
                  ( ( Xa
                    = ( some_P7363390416028606310at_nat @ A4 ) )
                 => ! [B3: product_prod_nat_nat] :
                      ( ( Xb
                        = ( some_P7363390416028606310at_nat @ B3 ) )
                     => ( ( Y
                          = ( some_P7363390416028606310at_nat @ ( X4 @ A4 @ B3 ) ) )
                       => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X4 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_2501_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X4: num > num > num,Xa: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X4 @ Xa @ Xb )
        = Y )
     => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X4 @ ( produc8585076106096196333on_num @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_num )
           => ( ( Y = none_num )
             => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X4 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
         => ( ! [V2: num] :
                ( ( Xa
                  = ( some_num @ V2 ) )
               => ( ( Xb = none_num )
                 => ( ( Y = none_num )
                   => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X4 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
           => ~ ! [A4: num] :
                  ( ( Xa
                    = ( some_num @ A4 ) )
                 => ! [B3: num] :
                      ( ( Xb
                        = ( some_num @ B3 ) )
                     => ( ( Y
                          = ( some_num @ ( X4 @ A4 @ B3 ) ) )
                       => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X4 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_2502_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X4: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X4 @ Xa @ Xb )
        = Y )
     => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X4 @ ( produc5098337634421038937on_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = none_nat )
           => ( ( Y = none_nat )
             => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X4 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
         => ( ! [V2: nat] :
                ( ( Xa
                  = ( some_nat @ V2 ) )
               => ( ( Xb = none_nat )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X4 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
           => ~ ! [A4: nat] :
                  ( ( Xa
                    = ( some_nat @ A4 ) )
                 => ! [B3: nat] :
                      ( ( Xb
                        = ( some_nat @ B3 ) )
                     => ( ( Y
                          = ( some_nat @ ( X4 @ A4 @ B3 ) ) )
                       => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X4 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_2503_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X4 ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X4 ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_2504_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_2505_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2506_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2507_sum__squares__gt__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2508_sum__squares__gt__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2509_sum__squares__gt__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2510_sum__squares__le__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2511_sum__squares__le__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2512_sum__squares__le__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2513_in__measure,axiom,
    ! [X4: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_2514_in__measure,axiom,
    ! [X4: int,Y: int,F: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ ( measure_int @ F ) )
      = ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_2515_in__measure,axiom,
    ! [X4: code_integer,Y: code_integer,F: code_integer > nat] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Y ) @ ( measure_Code_integer @ F ) )
      = ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_2516_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X4 )
        = Y )
     => ( ! [A4: $o] :
            ( ? [B3: $o] :
                ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_2517_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X4 ) @ N ) ) ).

% delete_pres_valid
thf(fact_2518_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X4 ) @ Y )
        = ( ( X4 != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_2519_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X4: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X4 ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X4 ) ) ).

% vebt_inserti_refines
thf(fact_2520_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_2521_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_2522_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_2523_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_2524_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X4 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X4 ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_2525_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X4 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_2526_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X4 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_2527_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2528_finite__lists__length__le,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2529_finite__lists__length__le,axiom,
    ! [A3: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2530_finite__lists__length__le,axiom,
    ! [A3: set_real,N: nat] :
      ( ( finite_finite_real @ A3 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2531_finite__lists__length__le,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2532_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2533_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2534_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_2535_divides__aux__eq,axiom,
    ! [Q5: code_integer,R3: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q5 @ R3 ) )
      = ( R3 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_2536_divides__aux__eq,axiom,
    ! [Q5: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q5 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2537_divides__aux__eq,axiom,
    ! [Q5: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q5 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2538_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2539_finite__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A3 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2540_finite__lists__length__eq,axiom,
    ! [A3: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A3 )
              & ( ( size_s3445333598471063425nteger @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2541_finite__lists__length__eq,axiom,
    ! [A3: set_real,N: nat] :
      ( ( finite_finite_real @ A3 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 )
              & ( ( size_size_list_real @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2542_finite__lists__length__eq,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
              & ( ( size_size_list_o @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2543_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
              & ( ( size_size_list_nat @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2544_finite__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
              & ( ( size_size_list_int @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2545_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_2546_add__shift,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_2547_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_2548_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_2549_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_2550_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_2551_neq__if__length__neq,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
       != ( size_size_list_real @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_2552_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_2553_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_2554_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_2555_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( ( finite306553202115118035t_real @ M8 )
     => ? [N2: nat] :
        ! [X6: list_real] :
          ( ( member_list_real @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_real @ X6 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_2556_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( ( finite_finite_list_o @ M8 )
     => ? [N2: nat] :
        ! [X6: list_o] :
          ( ( member_list_o @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_2557_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M8 )
     => ? [N2: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_2558_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( ( finite3922522038869484883st_int @ M8 )
     => ? [N2: nat] :
        ! [X6: list_int] :
          ( ( member_list_int @ X6 @ M8 )
         => ( ord_less_nat @ ( size_size_list_int @ X6 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_2559_length__induct,axiom,
    ! [P: list_real > $o,Xs2: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2560_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2561_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2562_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2563_subset__code_I1_J,axiom,
    ! [Xs2: list_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B4 )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
           => ( member_set_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2564_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B4 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2565_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2566_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B4 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2567_subset__code_I1_J,axiom,
    ! [Xs2: list_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ B4 )
      = ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( member_o @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2568_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B4 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_2569_length__pos__if__in__set,axiom,
    ! [X4: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2570_length__pos__if__in__set,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2571_length__pos__if__in__set,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2572_length__pos__if__in__set,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2573_length__pos__if__in__set,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2574_length__pos__if__in__set,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2575_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_2576_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_2577_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_2578_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X4 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_2579_vebt__insert_Osimps_I1_J,axiom,
    ! [X4: nat,A: $o,B: $o] :
      ( ( ( X4 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X4 )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X4 != zero_zero_nat )
       => ( ( ( X4 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X4 )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X4 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X4 )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_2580_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X4 )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(3)
thf(fact_2581_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X4 )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(2)
thf(fact_2582_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_2583_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X4 )
       => ( ( ( X4
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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_2584_add__scale__eq__noteq,axiom,
    ! [R3: real,A: real,B: real,C: real,D: real] :
      ( ( R3 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R3 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2585_add__scale__eq__noteq,axiom,
    ! [R3: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R3 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R3 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2586_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2587_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C: int,D: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2588_add__scale__eq__noteq,axiom,
    ! [R3: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R3 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R3 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2589_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X4 )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_2590_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > nat > nat,A1: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A1 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
     => ( ! [F2: nat > nat > nat,A4: nat,B3: nat,Acc: nat] :
            ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B3 @ A4 )
               => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F2 @ A4 @ Acc ) ) )
             => ( P @ F2 @ A4 @ B3 @ Acc ) ) )
       => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_2591_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > num > num,A1: nat,A22: nat,A32: num,P: ( nat > num > num ) > nat > nat > num > $o] :
      ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ A0 @ ( produc1195630363706982562at_num @ A1 @ ( product_Pair_nat_num @ A22 @ A32 ) ) ) )
     => ( ! [F2: nat > num > num,A4: nat,B3: nat,Acc: num] :
            ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A4 @ ( product_Pair_nat_num @ B3 @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B3 @ A4 )
               => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F2 @ A4 @ Acc ) ) )
             => ( P @ F2 @ A4 @ B3 @ Acc ) ) )
       => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_2592_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_2593_maxbmo,axiom,
    ! [T: vEBT_VEBT,X4: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X4 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X4 ) ) ).

% maxbmo
thf(fact_2594_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X4 ) @ Y )
        = ( ( X4 != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_2595_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
        = ( vEBT_vebt_member @ T @ X4 ) ) ) ).

% both_member_options_equiv_member
thf(fact_2596_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X4 )
       => ( vEBT_vebt_member @ T @ X4 ) ) ) ).

% valid_member_both_member_options
thf(fact_2597_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X )
          | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_2598_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_2599_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_2600_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_2601_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_2602_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_2603_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_2604_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_2605_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_2606_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_2607_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2608_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2609_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2610_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2611_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
       != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2612_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X4: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X4 @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z ) @ ( times_times_real @ X4 @ Y ) ) )
      = ( ( W2 = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2613_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X4: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X4 @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z ) @ ( times_times_rat @ X4 @ Y ) ) )
      = ( ( W2 = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2614_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X4: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X4 @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X4 @ Y ) ) )
      = ( ( W2 = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2615_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X4: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X4 @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z ) @ ( times_times_int @ X4 @ Y ) ) )
      = ( ( W2 = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2616_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X4: complex,Z: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y ) @ ( times_times_complex @ X4 @ Z ) )
        = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z ) @ ( times_times_complex @ X4 @ Y ) ) )
      = ( ( W2 = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2617_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X4 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X4 )
       => ( ! [Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X4
                    = ( 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_2618_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X4 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X4 )
       => ( ( ( X4
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_2619_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ 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 @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_2620_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X4: nat > num > num,Xa: nat,Xb: nat,Xc: num,Y: num] :
      ( ( ( set_fo8365102181078989356at_num @ X4 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X4 @ ( produc1195630363706982562at_num @ Xa @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo8365102181078989356at_num @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X4 @ ( produc1195630363706982562at_num @ Xa @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_2621_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X4 @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X4 @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_2622_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > num > num,A: nat,B: nat,Acc2: num] :
      ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F @ ( produc1195630363706982562at_num @ A @ ( product_Pair_nat_num @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
            = ( set_fo8365102181078989356at_num @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_2623_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_2624_foldr__same__int,axiom,
    ! [Xs2: list_nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Y ) ) ) ) ).

% foldr_same_int
thf(fact_2625_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_2626_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X4 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_2627_inthall,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,N: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2628_inthall,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,N: nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( P @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2629_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2630_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2631_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2632_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2633_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2634_set__n__lists,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( set_list_VEBT_VEBT2 @ ( n_lists_VEBT_VEBT @ N @ Xs2 ) )
      = ( collec5608196760682091941T_VEBT
        @ ^ [Ys3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N )
            & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Ys3 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_2635_set__n__lists,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( set_list_real2 @ ( n_lists_real @ N @ Xs2 ) )
      = ( collect_list_real
        @ ^ [Ys3: list_real] :
            ( ( ( size_size_list_real @ Ys3 )
              = N )
            & ( ord_less_eq_set_real @ ( set_real2 @ Ys3 ) @ ( set_real2 @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_2636_set__n__lists,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( set_list_o2 @ ( n_lists_o @ N @ Xs2 ) )
      = ( collect_list_o
        @ ^ [Ys3: list_o] :
            ( ( ( size_size_list_o @ Ys3 )
              = N )
            & ( ord_less_eq_set_o @ ( set_o2 @ Ys3 ) @ ( set_o2 @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_2637_set__n__lists,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
      = ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( size_size_list_nat @ Ys3 )
              = N )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_2638_set__n__lists,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( set_list_int2 @ ( n_lists_int @ N @ Xs2 ) )
      = ( collect_list_int
        @ ^ [Ys3: list_int] :
            ( ( ( size_size_list_int @ Ys3 )
              = N )
            & ( ord_less_eq_set_int @ ( set_int2 @ Ys3 ) @ ( set_int2 @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_2639_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P @ A4 @ B3 )
          = ( P @ B3 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ A4 @ B3 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_2640_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_2641_foldr__mono,axiom,
    ! [Xs2: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ( ord_less_eq_nat @ C @ D )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).

% foldr_mono
thf(fact_2642_foldr__length,axiom,
    ! [L: list_real] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_real @ L ) ) ).

% foldr_length
thf(fact_2643_foldr__length,axiom,
    ! [L: list_o] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_o @ L ) ) ).

% foldr_length
thf(fact_2644_foldr__length,axiom,
    ! [L: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_nat @ L ) ) ).

% foldr_length
thf(fact_2645_foldr__length,axiom,
    ! [L: list_int] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_int @ L ) ) ).

% foldr_length
thf(fact_2646_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBTi,Z4: list_VEBT_VEBTi] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2647_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2648_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_real,Z4: list_real] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
             => ( ( nth_real @ Xs @ I3 )
                = ( nth_real @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2649_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2650_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2651_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2652_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: vEBT_VEBTi] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2653_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: vEBT_VEBT] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2654_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: real] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_real] :
            ( ( ( size_size_list_real @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_real @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2655_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: $o] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2656_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: nat] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2657_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X8: int] : ( P @ I3 @ X8 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2658_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
              = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2659_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2660_nth__equalityI,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
           => ( ( nth_real @ Xs2 @ I2 )
              = ( nth_real @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2661_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2662_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2663_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_2664_obtain__list__from__elements,axiom,
    ! [N: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_VEBT_VEBTi @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2665_obtain__list__from__elements,axiom,
    ! [N: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_VEBT_VEBT @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2666_obtain__list__from__elements,axiom,
    ! [N: nat,P: real > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: real] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_real] :
            ( ( ( size_size_list_real @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_real @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2667_obtain__list__from__elements,axiom,
    ! [N: nat,P: $o > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: $o] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_o] :
            ( ( ( size_size_list_o @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_o @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2668_obtain__list__from__elements,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: nat] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_nat] :
            ( ( ( size_size_list_nat @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_nat @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2669_obtain__list__from__elements,axiom,
    ! [N: nat,P: int > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ? [Li: int] : ( P @ Li @ I2 ) )
     => ~ ! [L2: list_int] :
            ( ( ( size_size_list_int @ L2 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_int @ L2 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_2670_length__n__lists__elem,axiom,
    ! [Ys: list_real,N: nat,Xs2: list_real] :
      ( ( member_list_real @ Ys @ ( set_list_real2 @ ( n_lists_real @ N @ Xs2 ) ) )
     => ( ( size_size_list_real @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_2671_length__n__lists__elem,axiom,
    ! [Ys: list_o,N: nat,Xs2: list_o] :
      ( ( member_list_o @ Ys @ ( set_list_o2 @ ( n_lists_o @ N @ Xs2 ) ) )
     => ( ( size_size_list_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_2672_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs2: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_2673_length__n__lists__elem,axiom,
    ! [Ys: list_int,N: nat,Xs2: list_int] :
      ( ( member_list_int @ Ys @ ( set_list_int2 @ ( n_lists_int @ N @ Xs2 ) ) )
     => ( ( size_size_list_int @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_2674_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N ) @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2675_nth__mem,axiom,
    ! [N: nat,Xs2: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs2 @ N ) @ ( set_set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2676_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2677_nth__mem,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2678_nth__mem,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2679_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2680_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2681_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2682_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2683_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_real,P: real > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2684_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2685_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2686_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2687_in__set__conv__nth,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2688_in__set__conv__nth,axiom,
    ! [X4: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2689_in__set__conv__nth,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2690_in__set__conv__nth,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2691_in__set__conv__nth,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2692_in__set__conv__nth,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2693_in__set__conv__nth,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_2694_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X4: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2695_all__nth__imp__all__set,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,X4: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
         => ( P @ ( nth_set_nat @ Xs2 @ I2 ) ) )
     => ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2696_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2697_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X4: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2698_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X4: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2699_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X4: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2700_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X4: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_2701_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2702_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2703_all__set__conv__all__nth,axiom,
    ! [Xs2: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
           => ( P @ ( nth_real @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2704_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2705_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2706_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2707_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
           => ( P @ ( nth_VEBT_VEBTi @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2708_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
           => ( P @ ( nth_VEBT_VEBT @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2709_all__set__conv__nth,axiom,
    ! [L: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) )
           => ( P @ ( nth_real @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2710_all__set__conv__nth,axiom,
    ! [L: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ L ) )
           => ( P @ ( nth_o @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2711_all__set__conv__nth,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
           => ( P @ ( nth_nat @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2712_all__set__conv__nth,axiom,
    ! [L: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
           => ( P @ ( nth_int @ L @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_2713_size__list__estimation,axiom,
    ! [X4: int,Xs2: list_int,Y: nat,F: int > nat] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_int @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2714_size__list__estimation,axiom,
    ! [X4: set_nat,Xs2: list_set_nat,Y: nat,F: set_nat > nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_set_nat @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2715_size__list__estimation,axiom,
    ! [X4: nat,Xs2: list_nat,Y: nat,F: nat > nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2716_size__list__estimation,axiom,
    ! [X4: real,Xs2: list_real,Y: nat,F: real > nat] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_real @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2717_size__list__estimation,axiom,
    ! [X4: $o,Xs2: list_o,Y: nat,F: $o > nat] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_o @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2718_size__list__estimation,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y: nat,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( ord_less_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_nat @ Y @ ( size_list_VEBT_VEBT @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_2719_size__list__pointwise,axiom,
    ! [Xs2: list_int,F: int > nat,G: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_int @ F @ Xs2 ) @ ( size_list_int @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2720_size__list__pointwise,axiom,
    ! [Xs2: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_set_nat @ F @ Xs2 ) @ ( size_list_set_nat @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2721_size__list__pointwise,axiom,
    ! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs2 ) @ ( size_list_nat @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2722_size__list__pointwise,axiom,
    ! [Xs2: list_real,F: real > nat,G: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_real @ F @ Xs2 ) @ ( size_list_real @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2723_size__list__pointwise,axiom,
    ! [Xs2: list_o,F: $o > nat,G: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_o @ F @ Xs2 ) @ ( size_list_o @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2724_size__list__pointwise,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_nat @ ( size_list_VEBT_VEBT @ F @ Xs2 ) @ ( size_list_VEBT_VEBT @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_2725_size__list__estimation_H,axiom,
    ! [X4: int,Xs2: list_int,Y: nat,F: int > nat] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_int @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2726_size__list__estimation_H,axiom,
    ! [X4: set_nat,Xs2: list_set_nat,Y: nat,F: set_nat > nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_set_nat @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2727_size__list__estimation_H,axiom,
    ! [X4: nat,Xs2: list_nat,Y: nat,F: nat > nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2728_size__list__estimation_H,axiom,
    ! [X4: real,Xs2: list_real,Y: nat,F: real > nat] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_real @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2729_size__list__estimation_H,axiom,
    ! [X4: $o,Xs2: list_o,Y: nat,F: $o > nat] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_o @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2730_size__list__estimation_H,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y: nat,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X4 ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_VEBT_VEBT @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_2731_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,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_2732_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldr_length_aux
thf(fact_2733_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldr_length_aux
thf(fact_2734_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldr_length_aux
thf(fact_2735_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_int @ L ) ) ) ).

% foldr_length_aux
thf(fact_2736_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,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_2737_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_2738_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F4: nat > nat > nat,A2: nat,B2: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A2 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F4 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F4 @ A2 @ Acc3 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_2739_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,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_2740_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_2741_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_2742_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_2743_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 ) )
        @ ^ [R2: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R2
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_2744_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 ) )
        @ ^ [R2: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R2
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_2745_foldr__zero,axiom,
    ! [Xs2: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ D ) @ D ) ) ) ).

% foldr_zero
thf(fact_2746_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_2747_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2748_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_real,F: real > nat,Bound: nat,I: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_real_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2749_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_o,F: $o > nat,Bound: nat,I: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_o_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2750_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_nat,F: nat > nat,Bound: nat,I: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2751_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_int,F: int > nat,Bound: nat,I: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_int_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2752_length__mul__elem,axiom,
    ! [Xs2: list_list_real,N: nat] :
      ( ! [X3: list_real] :
          ( ( member_list_real @ X3 @ ( set_list_real2 @ Xs2 ) )
         => ( ( size_size_list_real @ X3 )
            = N ) )
     => ( ( size_size_list_real @ ( concat_real @ Xs2 ) )
        = ( times_times_nat @ ( size_s6660260683639930848t_real @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2753_length__mul__elem,axiom,
    ! [Xs2: list_list_o,N: nat] :
      ( ! [X3: list_o] :
          ( ( member_list_o @ X3 @ ( set_list_o2 @ Xs2 ) )
         => ( ( size_size_list_o @ X3 )
            = N ) )
     => ( ( size_size_list_o @ ( concat_o @ Xs2 ) )
        = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2754_length__mul__elem,axiom,
    ! [Xs2: list_list_nat,N: nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
         => ( ( size_size_list_nat @ X3 )
            = N ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs2 ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2755_length__mul__elem,axiom,
    ! [Xs2: list_list_int,N: nat] :
      ( ! [X3: list_int] :
          ( ( member_list_int @ X3 @ ( set_list_int2 @ Xs2 ) )
         => ( ( size_size_list_int @ X3 )
            = N ) )
     => ( ( size_size_list_int @ ( concat_int @ Xs2 ) )
        = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2756_ran__nth__set__encoding__conv,axiom,
    ! [L: list_VEBT_VEBTi] :
      ( ( ran_nat_VEBT_VEBTi
        @ ^ [I3: nat] : ( if_option_VEBT_VEBTi @ ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) ) @ ( some_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L @ I3 ) ) @ none_VEBT_VEBTi ) )
      = ( set_VEBT_VEBTi2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2757_ran__nth__set__encoding__conv,axiom,
    ! [L: list_VEBT_VEBT] :
      ( ( ran_nat_VEBT_VEBT
        @ ^ [I3: nat] : ( if_option_VEBT_VEBT @ ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) ) @ ( some_VEBT_VEBT @ ( nth_VEBT_VEBT @ L @ I3 ) ) @ none_VEBT_VEBT ) )
      = ( set_VEBT_VEBT2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2758_ran__nth__set__encoding__conv,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ( ( ran_na2114640787166747904at_nat
        @ ^ [I3: nat] : ( if_opt6109864365331422477at_nat @ ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ L ) ) @ ( some_P7363390416028606310at_nat @ ( nth_Pr7617993195940197384at_nat @ L @ I3 ) ) @ none_P5556105721700978146at_nat ) )
      = ( set_Pr5648618587558075414at_nat @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2759_ran__nth__set__encoding__conv,axiom,
    ! [L: list_num] :
      ( ( ran_nat_num
        @ ^ [I3: nat] : ( if_option_num @ ( ord_less_nat @ I3 @ ( size_size_list_num @ L ) ) @ ( some_num @ ( nth_num @ L @ I3 ) ) @ none_num ) )
      = ( set_num2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2760_ran__nth__set__encoding__conv,axiom,
    ! [L: list_real] :
      ( ( ran_nat_real
        @ ^ [I3: nat] : ( if_option_real @ ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) ) @ ( some_real @ ( nth_real @ L @ I3 ) ) @ none_real ) )
      = ( set_real2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2761_ran__nth__set__encoding__conv,axiom,
    ! [L: list_o] :
      ( ( ran_nat_o
        @ ^ [I3: nat] : ( if_option_o @ ( ord_less_nat @ I3 @ ( size_size_list_o @ L ) ) @ ( some_o @ ( nth_o @ L @ I3 ) ) @ none_o ) )
      = ( set_o2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2762_ran__nth__set__encoding__conv,axiom,
    ! [L: list_nat] :
      ( ( ran_nat_nat
        @ ^ [I3: nat] : ( if_option_nat @ ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) ) @ ( some_nat @ ( nth_nat @ L @ I3 ) ) @ none_nat ) )
      = ( set_nat2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2763_ran__nth__set__encoding__conv,axiom,
    ! [L: list_int] :
      ( ( ran_nat_int
        @ ^ [I3: nat] : ( if_option_int @ ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) ) @ ( some_int @ ( nth_int @ L @ I3 ) ) @ none_int ) )
      = ( set_int2 @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_2764_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_o,X4: $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 ) @ X4 )
             => ( ( ord_less_o @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_o @ X4 @ ( nth_o @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2765_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_real,X4: 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 ) @ X4 )
             => ( ( ord_less_real @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_real @ X4 @ ( nth_real @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2766_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_rat,X4: 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 ) @ X4 )
             => ( ( ord_less_rat @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_rat @ X4 @ ( nth_rat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2767_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,X4: 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 ) @ X4 )
             => ( ( ord_less_num @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_num @ X4 @ ( nth_num @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2768_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,X4: 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 ) @ X4 )
             => ( ( ord_less_nat @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_nat @ X4 @ ( nth_nat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2769_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,X4: 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 ) @ X4 )
             => ( ( ord_less_int @ X4 @ ( 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 ) @ X4 )
                         => ~ ( ord_less_int @ X4 @ ( nth_int @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_2770_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBTi,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_Pr3244165891152107629_VEBTi @ ( enumerate_VEBT_VEBTi @ N @ Xs2 ) @ M )
        = ( produc2649746096677893406_VEBTi @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBTi @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2771_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs2 ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2772_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_num,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_num @ Xs2 ) )
     => ( ( nth_Pr8326237132889035090at_num @ ( enumerate_num @ N @ Xs2 ) @ M )
        = ( product_Pair_nat_num @ ( plus_plus_nat @ N @ M ) @ ( nth_num @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2773_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_real,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_Pr7767817059697521252t_real @ ( enumerate_real @ N @ Xs2 ) @ M )
        = ( produc7837566107596912789t_real @ ( plus_plus_nat @ N @ M ) @ ( nth_real @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2774_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_o,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs2 ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2775_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs2 ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2776_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_int,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs2 ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2777_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs: list_nat] : Xs ) ) ).

% map_ident
thf(fact_2778_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2779_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_2780_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_2781_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_2782_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_2783_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_2784_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_2785_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_2786_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_2787_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_2788_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_2789_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_2790_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_2791_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_2792_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_2793_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_2794_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_2795_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_2796_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_2797_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_2798_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_2799_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_2800_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_2801_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2802_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2803_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2804_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_2805_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_2806_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_2807_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_2808_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_2809_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2810_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2811_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2812_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_2813_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2814_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2815_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_2816_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2817_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2818_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_2819_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_2820_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_2821_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_2822_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_2823_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_2824_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_2825_length__map,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_2826_length__map,axiom,
    ! [F: real > real,Xs2: list_real] :
      ( ( size_size_list_real @ ( map_real_real @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_2827_length__map,axiom,
    ! [F: $o > real,Xs2: list_o] :
      ( ( size_size_list_real @ ( map_o_real @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_2828_length__map,axiom,
    ! [F: nat > real,Xs2: list_nat] :
      ( ( size_size_list_real @ ( map_nat_real @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_2829_length__map,axiom,
    ! [F: int > real,Xs2: list_int] :
      ( ( size_size_list_real @ ( map_int_real @ F @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_map
thf(fact_2830_length__map,axiom,
    ! [F: real > $o,Xs2: list_real] :
      ( ( size_size_list_o @ ( map_real_o @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_2831_length__map,axiom,
    ! [F: $o > $o,Xs2: list_o] :
      ( ( size_size_list_o @ ( map_o_o @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_2832_length__map,axiom,
    ! [F: nat > $o,Xs2: list_nat] :
      ( ( size_size_list_o @ ( map_nat_o @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_2833_length__map,axiom,
    ! [F: int > $o,Xs2: list_int] :
      ( ( size_size_list_o @ ( map_int_o @ F @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_map
thf(fact_2834_length__map,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_2835_length__enumerate,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( size_s7910714270633306959t_real @ ( enumerate_real @ N @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_enumerate
thf(fact_2836_length__enumerate,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_enumerate
thf(fact_2837_length__enumerate,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_2838_length__enumerate,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_enumerate
thf(fact_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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_2851_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_2852_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_2853_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_2854_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_2855_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_2866_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_2867_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_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > nat] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBTi_nat @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2876_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2877_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2878_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2879_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2880_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > int] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_int @ ( map_VEBT_VEBTi_int @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2881_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_int @ ( map_VEBT_VEBT_int @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2882_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2883_nth__map,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2884_nth__map,axiom,
    ! [N: nat,Xs2: list_real,F: real > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_nat @ ( map_real_nat @ F @ Xs2 ) @ N )
        = ( F @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% nth_map
thf(fact_2885_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_2886_sorted__map,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
      = ( sorted_wrt_VEBT_VEBT
        @ ^ [X: vEBT_VEBT,Y4: vEBT_VEBT] : ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2887_sorted__map,axiom,
    ! [F: nat > $o,Xs2: list_nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( map_nat_o @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( ord_less_eq_o @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2888_sorted__map,axiom,
    ! [F: nat > rat,Xs2: list_nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( map_nat_rat @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2889_sorted__map,axiom,
    ! [F: int > rat,Xs2: list_int] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( map_int_rat @ F @ Xs2 ) )
      = ( sorted_wrt_int
        @ ^ [X: int,Y4: int] : ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2890_sorted__map,axiom,
    ! [F: nat > num,Xs2: list_nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( map_nat_num @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2891_sorted__map,axiom,
    ! [F: int > num,Xs2: list_int] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( map_int_num @ F @ Xs2 ) )
      = ( sorted_wrt_int
        @ ^ [X: int,Y4: int] : ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2892_sorted__map,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) )
      = ( sorted_wrt_VEBT_VEBT
        @ ^ [X: vEBT_VEBT,Y4: vEBT_VEBT] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2893_sorted__map,axiom,
    ! [F: nat > nat,Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2894_sorted__map,axiom,
    ! [F: int > nat,Xs2: list_int] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_int_nat @ F @ Xs2 ) )
      = ( sorted_wrt_int
        @ ^ [X: int,Y4: int] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2895_sorted__map,axiom,
    ! [F: nat > int,Xs2: list_nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_nat_int @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_map
thf(fact_2896_sorted__wrt__true,axiom,
    ! [Xs2: list_nat] :
      ( sorted_wrt_nat
      @ ^ [Uu3: nat,Uv3: nat] : $true
      @ Xs2 ) ).

% sorted_wrt_true
thf(fact_2897_sorted__wrt__true,axiom,
    ! [Xs2: list_int] :
      ( sorted_wrt_int
      @ ^ [Uu3: int,Uv3: int] : $true
      @ Xs2 ) ).

% sorted_wrt_true
thf(fact_2898_sorted__wrt__map,axiom,
    ! [R: real > real > $o,F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_real @ R @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
      = ( sorted_wrt_VEBT_VEBT
        @ ^ [X: vEBT_VEBT,Y4: vEBT_VEBT] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2899_sorted__wrt__map,axiom,
    ! [R: $o > $o > $o,F: product_prod_o_o > $o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( sorted_wrt_o @ R @ ( map_Pr7541730621154948341_o_o_o @ F @ Xs2 ) )
      = ( sorted4470838345478584340od_o_o
        @ ^ [X: product_prod_o_o,Y4: product_prod_o_o] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2900_sorted__wrt__map,axiom,
    ! [R: $o > $o > $o,F: nat > $o,Xs2: list_nat] :
      ( ( sorted_wrt_o @ R @ ( map_nat_o @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2901_sorted__wrt__map,axiom,
    ! [R: nat > nat > $o,F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_nat @ R @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) )
      = ( sorted_wrt_VEBT_VEBT
        @ ^ [X: vEBT_VEBT,Y4: vEBT_VEBT] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2902_sorted__wrt__map,axiom,
    ! [R: nat > nat > $o,F: nat > nat,Xs2: list_nat] :
      ( ( sorted_wrt_nat @ R @ ( map_nat_nat @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2903_sorted__wrt__map,axiom,
    ! [R: nat > nat > $o,F: int > nat,Xs2: list_int] :
      ( ( sorted_wrt_nat @ R @ ( map_int_nat @ F @ Xs2 ) )
      = ( sorted_wrt_int
        @ ^ [X: int,Y4: int] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2904_sorted__wrt__map,axiom,
    ! [R: int > int > $o,F: nat > int,Xs2: list_nat] :
      ( ( sorted_wrt_int @ R @ ( map_nat_int @ F @ Xs2 ) )
      = ( sorted_wrt_nat
        @ ^ [X: nat,Y4: nat] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2905_sorted__wrt__map,axiom,
    ! [R: int > int > $o,F: int > int,Xs2: list_int] :
      ( ( sorted_wrt_int @ R @ ( map_int_int @ F @ Xs2 ) )
      = ( sorted_wrt_int
        @ ^ [X: int,Y4: int] : ( R @ ( F @ X ) @ ( F @ Y4 ) )
        @ Xs2 ) ) ).

% sorted_wrt_map
thf(fact_2906_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_2907_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_2908_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2909_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2910_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2911_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2912_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2913_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2914_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_2915_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2916_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2917_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: real > nat,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_real_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2918_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: real > real,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_real_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2919_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: $o > nat,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_o_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2920_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: $o > real,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_o_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2921_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: nat > nat,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2922_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: nat > real,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_nat_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2923_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: int > nat,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_int_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2924_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: int > real,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_int_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2925_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: complex,Z4: complex] : ( Y6 = Z4 ) )
    = ( ^ [A2: complex,B2: complex] :
          ( ( minus_minus_complex @ A2 @ B2 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2926_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
    = ( ^ [A2: real,B2: real] :
          ( ( minus_minus_real @ A2 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2927_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
    = ( ^ [A2: rat,B2: rat] :
          ( ( minus_minus_rat @ A2 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2928_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2929_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2930_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2931_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2932_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2933_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2934_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_2935_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2936_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2937_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_2938_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2939_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2940_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_2941_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2942_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2943_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_2944_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2945_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2946_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2947_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2948_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2949_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_2950_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2951_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2952_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_2953_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_2954_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_2955_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_2956_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_2957_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2958_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2959_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2960_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_2961_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2962_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2963_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2964_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_2965_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_2966_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_2967_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2968_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2969_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_2970_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_2971_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_2972_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_2973_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_2974_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_2975_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_2976_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_2977_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_2978_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_2979_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A3 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2980_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2981_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_2982_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X6: real,K4: real] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2983_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2984_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2985_inf__period_I1_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X3: complex,K2: complex] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
     => ( ! [X3: complex,K2: complex] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
       => ! [X6: complex,K4: complex] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2986_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X6: real,K4: real] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2987_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2988_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2989_inf__period_I2_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X3: complex,K2: complex] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
     => ( ! [X3: complex,K2: complex] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
       => ! [X6: complex,K4: complex] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2990_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2991_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2992_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2993_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2994_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2995_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2996_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2997_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2998_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2999_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3000_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3001_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3002_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3003_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3004_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3005_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3006_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3007_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3008_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_3009_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_3010_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_3011_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_3012_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_3013_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_3014_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_3015_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_3016_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_3017_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_3018_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_3019_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_3020_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_3021_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_3022_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_3023_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_3024_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_3025_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_3026_strict__sorted__imp__sorted,axiom,
    ! [Xs2: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ Xs2 )
     => ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).

% strict_sorted_imp_sorted
thf(fact_3027_strict__sorted__imp__sorted,axiom,
    ! [Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ Xs2 )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 ) ) ).

% strict_sorted_imp_sorted
thf(fact_3028_strict__sorted__imp__sorted,axiom,
    ! [Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs2 )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs2 ) ) ).

% strict_sorted_imp_sorted
thf(fact_3029_strict__sorted__imp__sorted,axiom,
    ! [Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).

% strict_sorted_imp_sorted
thf(fact_3030_strict__sorted__imp__sorted,axiom,
    ! [Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs2 )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).

% strict_sorted_imp_sorted
thf(fact_3031_strict__sorted__equal,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_o @ Xs2 )
     => ( ( sorted_wrt_o @ ord_less_o @ Ys )
       => ( ( ( set_o2 @ Ys )
            = ( set_o2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3032_strict__sorted__equal,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ Xs2 )
     => ( ( sorted_wrt_real @ ord_less_real @ Ys )
       => ( ( ( set_real2 @ Ys )
            = ( set_real2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3033_strict__sorted__equal,axiom,
    ! [Xs2: list_rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ Xs2 )
     => ( ( sorted_wrt_rat @ ord_less_rat @ Ys )
       => ( ( ( set_rat2 @ Ys )
            = ( set_rat2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3034_strict__sorted__equal,axiom,
    ! [Xs2: list_num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs2 )
     => ( ( sorted_wrt_num @ ord_less_num @ Ys )
       => ( ( ( set_num2 @ Ys )
            = ( set_num2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3035_strict__sorted__equal,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
     => ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
       => ( ( ( set_nat2 @ Ys )
            = ( set_nat2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3036_strict__sorted__equal,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs2 )
     => ( ( sorted_wrt_int @ ord_less_int @ Ys )
       => ( ( ( set_int2 @ Ys )
            = ( set_int2 @ Xs2 ) )
         => ( Ys = Xs2 ) ) ) ) ).

% strict_sorted_equal
thf(fact_3037_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3038_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3039_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3040_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3041_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3042_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3052_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3053_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3054_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3055_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3056_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3057_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_3058_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_3059_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3060_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3061_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3062_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3063_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3064_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3065_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_3066_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3067_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_3068_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_3069_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_3070_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_3071_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3072_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3073_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3074_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3075_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3076_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3077_square__diff__square__factored,axiom,
    ! [X4: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_real @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3078_square__diff__square__factored,axiom,
    ! [X4: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( minus_minus_rat @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3079_square__diff__square__factored,axiom,
    ! [X4: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X4 @ Y ) @ ( minus_minus_int @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3080_square__diff__square__factored,axiom,
    ! [X4: complex,Y: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ ( times_times_complex @ Y @ Y ) )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ Y ) @ ( minus_minus_complex @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3081_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3082_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3083_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3084_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3085_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3086_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3087_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3088_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3089_mult__diff__mult,axiom,
    ! [X4: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3090_mult__diff__mult,axiom,
    ! [X4: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X4 @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3091_mult__diff__mult,axiom,
    ! [X4: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3092_mult__diff__mult,axiom,
    ! [X4: complex,Y: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ Y ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X4 @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3093_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_3094_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_3095_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_3096_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_3097_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_3098_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_3099_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_3100_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_3101_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_3102_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_3103_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_3104_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_3105_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_3106_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_3107_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_3108_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_3109_sorted__wrt01,axiom,
    ! [Xs2: list_real,P: real > real > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_real @ P @ Xs2 ) ) ).

% sorted_wrt01
thf(fact_3110_sorted__wrt01,axiom,
    ! [Xs2: list_o,P: $o > $o > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_o @ P @ Xs2 ) ) ).

% sorted_wrt01
thf(fact_3111_sorted__wrt01,axiom,
    ! [Xs2: list_nat,P: nat > nat > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_nat @ P @ Xs2 ) ) ).

% sorted_wrt01
thf(fact_3112_sorted__wrt01,axiom,
    ! [Xs2: list_int,P: int > int > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_int @ P @ Xs2 ) ) ).

% sorted_wrt01
thf(fact_3113_sorted__wrt__iff__nth__less,axiom,
    ( sorted9206477368072086664_VEBTi
    = ( ^ [P2: vEBT_VEBTi > vEBT_VEBTi > $o,Xs: list_VEBT_VEBTi] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( P2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3114_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_VEBT_VEBT
    = ( ^ [P2: vEBT_VEBT > vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3115_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_real
    = ( ^ [P2: real > real > $o,Xs: list_real] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs ) )
           => ( P2 @ ( nth_real @ Xs @ I3 ) @ ( nth_real @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3116_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_o
    = ( ^ [P2: $o > $o > $o,Xs: list_o] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
           => ( P2 @ ( nth_o @ Xs @ I3 ) @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3117_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P2: nat > nat > $o,Xs: list_nat] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
           => ( P2 @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3118_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P2: int > int > $o,Xs: list_int] :
        ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
           => ( P2 @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3119_sorted__wrt__nth__less,axiom,
    ! [P: vEBT_VEBTi > vEBT_VEBTi > $o,Xs2: list_VEBT_VEBTi,I: nat,J: nat] :
      ( ( sorted9206477368072086664_VEBTi @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3120_sorted__wrt__nth__less,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBT > $o,Xs2: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( sorted_wrt_VEBT_VEBT @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3121_sorted__wrt__nth__less,axiom,
    ! [P: real > real > $o,Xs2: list_real,I: nat,J: nat] :
      ( ( sorted_wrt_real @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I ) @ ( nth_real @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3122_sorted__wrt__nth__less,axiom,
    ! [P: $o > $o > $o,Xs2: list_o,I: nat,J: nat] :
      ( ( sorted_wrt_o @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I ) @ ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3123_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs2: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3124_sorted__wrt__nth__less,axiom,
    ! [P: int > int > $o,Xs2: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ P @ Xs2 )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I ) @ ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3125_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3126_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3127_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3128_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3129_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3130_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3131_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3132_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3133_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3134_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3135_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3136_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3137_square__diff__one__factored,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_3138_square__diff__one__factored,axiom,
    ! [X4: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_3139_square__diff__one__factored,axiom,
    ! [X4: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3140_square__diff__one__factored,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_3141_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_3142_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_3143_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_3144_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3145_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_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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_3152_diff__preserves__multiset,axiom,
    ! [M8: list_nat > nat,N7: list_nat > nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3153_diff__preserves__multiset,axiom,
    ! [M8: set_nat > nat,N7: set_nat > nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3154_diff__preserves__multiset,axiom,
    ! [M8: nat > nat,N7: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3155_diff__preserves__multiset,axiom,
    ! [M8: int > nat,N7: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3156_diff__preserves__multiset,axiom,
    ! [M8: complex > nat,N7: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3157_diff__preserves__multiset,axiom,
    ! [M8: code_integer > nat,N7: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N7 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3158_sorted01,axiom,
    ! [Xs2: list_real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).

% sorted01
thf(fact_3159_sorted01,axiom,
    ! [Xs2: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_o @ ord_less_eq_o @ Xs2 ) ) ).

% sorted01
thf(fact_3160_sorted01,axiom,
    ! [Xs2: list_rat] :
      ( ( ord_less_eq_nat @ ( size_size_list_rat @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 ) ) ).

% sorted01
thf(fact_3161_sorted01,axiom,
    ! [Xs2: list_num] :
      ( ( ord_less_eq_nat @ ( size_size_list_num @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs2 ) ) ).

% sorted01
thf(fact_3162_sorted01,axiom,
    ! [Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).

% sorted01
thf(fact_3163_sorted01,axiom,
    ! [Xs2: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).

% sorted01
thf(fact_3164_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
             => ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3165_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs2 ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs2 @ I3 ) @ ( nth_o @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3166_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs2 ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs2 @ I3 ) @ ( nth_rat @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3167_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs2 ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs2 @ I3 ) @ ( nth_num @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3168_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3169_sorted__iff__nth__mono__less,axiom,
    ! [Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3170_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_3171_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_3172_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_3173_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_3174_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_3175_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3176_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_3177_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_3178_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_real @ Xs2 ) )
           => ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3179_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_o @ Xs2 ) )
           => ( ord_less_eq_o @ ( nth_o @ Xs2 @ I3 ) @ ( nth_o @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3180_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_rat @ Xs2 ) )
           => ( ord_less_eq_rat @ ( nth_rat @ Xs2 @ I3 ) @ ( nth_rat @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3181_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_num @ Xs2 ) )
           => ( ord_less_eq_num @ ( nth_num @ Xs2 @ I3 ) @ ( nth_num @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3182_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3183_sorted__iff__nth__Suc,axiom,
    ! [Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_int @ Xs2 ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ ( suc @ I3 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3184_sorted__nth__mono,axiom,
    ! [Xs2: list_real,I: nat,J: nat] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
         => ( ord_less_eq_real @ ( nth_real @ Xs2 @ I ) @ ( nth_real @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3185_sorted__nth__mono,axiom,
    ! [Xs2: list_o,I: nat,J: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
         => ( ord_less_eq_o @ ( nth_o @ Xs2 @ I ) @ ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3186_sorted__nth__mono,axiom,
    ! [Xs2: list_rat,I: nat,J: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_rat @ Xs2 ) )
         => ( ord_less_eq_rat @ ( nth_rat @ Xs2 @ I ) @ ( nth_rat @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3187_sorted__nth__mono,axiom,
    ! [Xs2: list_num,I: nat,J: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs2 ) )
         => ( ord_less_eq_num @ ( nth_num @ Xs2 @ I ) @ ( nth_num @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3188_sorted__nth__mono,axiom,
    ! [Xs2: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3189_sorted__nth__mono,axiom,
    ! [Xs2: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs2 @ I ) @ ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3190_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs2 ) )
             => ( ord_less_eq_real @ ( nth_real @ Xs2 @ I3 ) @ ( nth_real @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3191_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs2 ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs2 @ I3 ) @ ( nth_o @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3192_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs2 ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs2 @ I3 ) @ ( nth_rat @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3193_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs2 ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs2 @ I3 ) @ ( nth_num @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3194_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3195_sorted__iff__nth__mono,axiom,
    ! [Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
      = ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs2 @ I3 ) @ ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3196_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_3197_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A3: real > real > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L1930518968523514909l_real @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3198_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_o,A3: real > $o > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L6234343332106409831real_o @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3199_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A3: real > nat > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L1446010312343316929al_nat @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3200_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_int,A3: real > int > assn] :
      ( ( ( size_size_list_real @ L )
       != ( size_size_list_int @ Li2 ) )
     => ( ( vEBT_L1443519841834266653al_int @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3201_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_real,A3: $o > real > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L4725278957065240257o_real @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3202_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_o,A3: $o > $o > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L7363604446928714179sn_o_o @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3203_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_nat,A3: $o > nat > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L4785011123346445925_o_nat @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3204_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_int,A3: $o > int > assn] :
      ( ( ( size_size_list_o @ L )
       != ( size_size_list_int @ Li2 ) )
     => ( ( vEBT_L4782520652837395649_o_int @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3205_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_real,A3: nat > real > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L6102073776069194049t_real @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3206_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li2: list_o,A3: nat > $o > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L7887682484454631235_nat_o @ A3 @ L @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_3207_repli__emp,axiom,
    ! [X4: heap_T2636463487746394924on_nat,A3: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_6480275734082232733on_nat @ one_one_assn @ ( vEBT_V792416675989592002on_nat @ N @ X4 ) @ ( vEBT_L8010285020845282001on_nat @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3208_repli__emp,axiom,
    ! [X4: heap_T2636463487746394924on_nat,A3: $o > option_nat > assn,Y: $o,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_6480275734082232733on_nat @ one_one_assn @ ( vEBT_V792416675989592002on_nat @ N @ X4 ) @ ( vEBT_L2956511777047245877on_nat @ A3 @ ( replicate_o @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3209_repli__emp,axiom,
    ! [X4: heap_Time_Heap_o,A3: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_hoare_triple_o @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_9089481587091695345list_o @ one_one_assn @ ( vEBT_V2326993469660664182atei_o @ N @ X4 ) @ ( vEBT_L7489408758114837031VEBT_o @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3210_repli__emp,axiom,
    ! [X4: heap_Time_Heap_o,A3: $o > $o > assn,Y: $o,N: nat] :
      ( ( hoare_hoare_triple_o @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_9089481587091695345list_o @ one_one_assn @ ( vEBT_V2326993469660664182atei_o @ N @ X4 ) @ ( vEBT_L7363604446928714179sn_o_o @ A3 @ ( replicate_o @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3211_repli__emp,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,A3: $o > vEBT_VEBTi > assn,Y: $o,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_3904069481286416050_VEBTi @ one_one_assn @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 ) @ ( vEBT_L3704169666673096010_VEBTi @ A3 @ ( replicate_o @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3212_repli__emp,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,A3: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_3904069481286416050_VEBTi @ one_one_assn @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 ) @ ( vEBT_L6296928887356842470_VEBTi @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3213_repli__emp,axiom,
    ! [X4: heap_Time_Heap_nat,A3: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_7964568885773372237st_nat @ one_one_assn @ ( vEBT_V7726092123322077554ei_nat @ N @ X4 ) @ ( vEBT_L8296926524756676353BT_nat @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3214_repli__emp,axiom,
    ! [X4: heap_Time_Heap_nat,A3: $o > nat > assn,Y: $o,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ one_one_assn @ X4 @ ( A3 @ Y ) )
     => ( hoare_7964568885773372237st_nat @ one_one_assn @ ( vEBT_V7726092123322077554ei_nat @ N @ X4 ) @ ( vEBT_L4785011123346445925_o_nat @ A3 @ ( replicate_o @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_3215_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_T2636463487746394924on_nat,A3: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ Q @ X4
        @ ^ [R2: option_nat] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_6480275734082232733on_nat @ Q @ ( vEBT_V792416675989592002on_nat @ N @ X4 )
        @ ^ [R2: list_option_nat] : ( times_times_assn @ Q @ ( vEBT_L8010285020845282001on_nat @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3216_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_T2636463487746394924on_nat,A3: $o > option_nat > assn,Y: $o,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ Q @ X4
        @ ^ [R2: option_nat] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_6480275734082232733on_nat @ Q @ ( vEBT_V792416675989592002on_nat @ N @ X4 )
        @ ^ [R2: list_option_nat] : ( times_times_assn @ Q @ ( vEBT_L2956511777047245877on_nat @ A3 @ ( replicate_o @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3217_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_Time_Heap_o,A3: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_hoare_triple_o @ Q @ X4
        @ ^ [R2: $o] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_9089481587091695345list_o @ Q @ ( vEBT_V2326993469660664182atei_o @ N @ X4 )
        @ ^ [R2: list_o] : ( times_times_assn @ Q @ ( vEBT_L7489408758114837031VEBT_o @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3218_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_Time_Heap_o,A3: $o > $o > assn,Y: $o,N: nat] :
      ( ( hoare_hoare_triple_o @ Q @ X4
        @ ^ [R2: $o] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_9089481587091695345list_o @ Q @ ( vEBT_V2326993469660664182atei_o @ N @ X4 )
        @ ^ [R2: list_o] : ( times_times_assn @ Q @ ( vEBT_L7363604446928714179sn_o_o @ A3 @ ( replicate_o @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3219_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_T8145700208782473153_VEBTi,A3: $o > vEBT_VEBTi > assn,Y: $o,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ Q @ X4
        @ ^ [R2: vEBT_VEBTi] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_3904069481286416050_VEBTi @ Q @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 )
        @ ^ [R2: list_VEBT_VEBTi] : ( times_times_assn @ Q @ ( vEBT_L3704169666673096010_VEBTi @ A3 @ ( replicate_o @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3220_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_T8145700208782473153_VEBTi,A3: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ Q @ X4
        @ ^ [R2: vEBT_VEBTi] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_3904069481286416050_VEBTi @ Q @ ( vEBT_V1859673955506687831_VEBTi @ N @ X4 )
        @ ^ [R2: list_VEBT_VEBTi] : ( times_times_assn @ Q @ ( vEBT_L6296928887356842470_VEBTi @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3221_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_Time_Heap_nat,A3: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ Q @ X4
        @ ^ [R2: nat] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_7964568885773372237st_nat @ Q @ ( vEBT_V7726092123322077554ei_nat @ N @ X4 )
        @ ^ [R2: list_nat] : ( times_times_assn @ Q @ ( vEBT_L8296926524756676353BT_nat @ A3 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3222_repli__cons__repl,axiom,
    ! [Q: assn,X4: heap_Time_Heap_nat,A3: $o > nat > assn,Y: $o,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ Q @ X4
        @ ^ [R2: nat] : ( times_times_assn @ Q @ ( A3 @ Y @ R2 ) ) )
     => ( hoare_7964568885773372237st_nat @ Q @ ( vEBT_V7726092123322077554ei_nat @ N @ X4 )
        @ ^ [R2: list_nat] : ( times_times_assn @ Q @ ( vEBT_L4785011123346445925_o_nat @ A3 @ ( replicate_o @ N @ Y ) @ R2 ) ) ) ) ).

% repli_cons_repl
thf(fact_3223_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X4 )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( 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_3224_diff__shunt__var,axiom,
    ! [X4: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X4 @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X4 @ Y ) ) ).

% diff_shunt_var
thf(fact_3225_diff__shunt__var,axiom,
    ! [X4: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X4 @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X4 @ Y ) ) ).

% diff_shunt_var
thf(fact_3226_diff__shunt__var,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X4 @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X4 @ Y ) ) ).

% diff_shunt_var
thf(fact_3227_diff__shunt__var,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X4 @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% diff_shunt_var
thf(fact_3228_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_VEBT_VEBTi,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_VEBT_VEBTi @ ( slice_VEBT_VEBTi @ From @ To @ Xs2 ) @ I )
            = ( nth_VEBT_VEBTi @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3229_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_VEBT_VEBT,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_VEBT_VEBT @ ( slice_VEBT_VEBT @ From @ To @ Xs2 ) @ I )
            = ( nth_VEBT_VEBT @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3230_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_real,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_real @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_real @ ( slice_real @ From @ To @ Xs2 ) @ I )
            = ( nth_real @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3231_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_o,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_o @ ( slice_o @ From @ To @ Xs2 ) @ I )
            = ( nth_o @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3232_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_nat,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_nat @ ( slice_nat @ From @ To @ Xs2 ) @ I )
            = ( nth_nat @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3233_slice__nth,axiom,
    ! [From: nat,To: nat,Xs2: list_int,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs2 ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_int @ ( slice_int @ From @ To @ Xs2 ) @ I )
            = ( nth_int @ Xs2 @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3234_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_3235_intind,axiom,
    ! [I: nat,N: nat,P: nat > $o,X4: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X4 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N @ X4 ) @ I ) ) ) ) ).

% intind
thf(fact_3236_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBTi > $o,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X4 )
       => ( P @ ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N @ X4 ) @ I ) ) ) ) ).

% intind
thf(fact_3237_intind,axiom,
    ! [I: nat,N: nat,P: int > $o,X4: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X4 )
       => ( P @ ( nth_int @ ( replicate_int @ N @ X4 ) @ I ) ) ) ) ).

% intind
thf(fact_3238_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X4 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X4 ) @ I ) ) ) ) ).

% intind
thf(fact_3239_intind,axiom,
    ! [I: nat,N: nat,P: $o > $o,X4: $o] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X4 )
       => ( P @ ( nth_o @ ( replicate_o @ N @ X4 ) @ I ) ) ) ) ).

% intind
thf(fact_3240_Diff__cancel,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ A3 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3241_Diff__cancel,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ A3 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_3242_Diff__cancel,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ A3 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3243_Diff__cancel,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ A3 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3244_empty__Diff,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3245_empty__Diff,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_3246_empty__Diff,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3247_empty__Diff,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3248_Diff__empty,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
      = A3 ) ).

% Diff_empty
thf(fact_3249_Diff__empty,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Diff_empty
thf(fact_3250_Diff__empty,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Diff_empty
thf(fact_3251_Diff__empty,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Diff_empty
thf(fact_3252_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ( minus_minus_set_real @ A3 @ B4 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_3253_Diff__eq__empty__iff,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B4 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_3254_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_3255_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_3256_replicate__eq__replicate,axiom,
    ! [M: nat,X4: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X4 )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X4 = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_3257_replicate__eq__replicate,axiom,
    ! [M: nat,X4: $o,N: nat,Y: $o] :
      ( ( ( replicate_o @ M @ X4 )
        = ( replicate_o @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X4 = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_3258_length__replicate,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X4 ) )
      = N ) ).

% length_replicate
thf(fact_3259_length__replicate,axiom,
    ! [N: nat,X4: real] :
      ( ( size_size_list_real @ ( replicate_real @ N @ X4 ) )
      = N ) ).

% length_replicate
thf(fact_3260_length__replicate,axiom,
    ! [N: nat,X4: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X4 ) )
      = N ) ).

% length_replicate
thf(fact_3261_length__replicate,axiom,
    ! [N: nat,X4: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X4 ) )
      = N ) ).

% length_replicate
thf(fact_3262_length__replicate,axiom,
    ! [N: nat,X4: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X4 ) )
      = N ) ).

% length_replicate
thf(fact_3263_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3264_Ball__set__replicate,axiom,
    ! [N: nat,A: real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3265_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3266_Ball__set__replicate,axiom,
    ! [N: nat,A: $o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ ( replicate_o @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3267_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3268_Bex__set__replicate,axiom,
    ! [N: nat,A: real,P: real > $o] :
      ( ( ? [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3269_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3270_Bex__set__replicate,axiom,
    ! [N: nat,A: $o,P: $o > $o] :
      ( ( ? [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ ( replicate_o @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3271_in__set__replicate,axiom,
    ! [X4: int,N: nat,Y: int] :
      ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3272_in__set__replicate,axiom,
    ! [X4: set_nat,N: nat,Y: set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3273_in__set__replicate,axiom,
    ! [X4: nat,N: nat,Y: nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3274_in__set__replicate,axiom,
    ! [X4: real,N: nat,Y: real] :
      ( ( member_real @ X4 @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3275_in__set__replicate,axiom,
    ! [X4: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3276_in__set__replicate,axiom,
    ! [X4: $o,N: nat,Y: $o] :
      ( ( member_o @ X4 @ ( set_o2 @ ( replicate_o @ N @ Y ) ) )
      = ( ( X4 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3277_nth__replicate,axiom,
    ! [I: nat,N: nat,X4: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X4 ) @ I )
        = X4 ) ) ).

% nth_replicate
thf(fact_3278_nth__replicate,axiom,
    ! [I: nat,N: nat,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N @ X4 ) @ I )
        = X4 ) ) ).

% nth_replicate
thf(fact_3279_nth__replicate,axiom,
    ! [I: nat,N: nat,X4: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( replicate_int @ N @ X4 ) @ I )
        = X4 ) ) ).

% nth_replicate
thf(fact_3280_nth__replicate,axiom,
    ! [I: nat,N: nat,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X4 ) @ I )
        = X4 ) ) ).

% nth_replicate
thf(fact_3281_nth__replicate,axiom,
    ! [I: nat,N: nat,X4: $o] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_o @ ( replicate_o @ N @ X4 ) @ I )
        = X4 ) ) ).

% nth_replicate
thf(fact_3282_slice__complete,axiom,
    ! [Xs2: list_real] :
      ( ( slice_real @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) @ Xs2 )
      = Xs2 ) ).

% slice_complete
thf(fact_3283_slice__complete,axiom,
    ! [Xs2: list_o] :
      ( ( slice_o @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) @ Xs2 )
      = Xs2 ) ).

% slice_complete
thf(fact_3284_slice__complete,axiom,
    ! [Xs2: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) @ Xs2 )
      = Xs2 ) ).

% slice_complete
thf(fact_3285_slice__complete,axiom,
    ! [Xs2: list_int] :
      ( ( slice_int @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) @ Xs2 )
      = Xs2 ) ).

% slice_complete
thf(fact_3286_slice__len,axiom,
    ! [From: nat,To: nat,Xs2: list_real] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_real @ Xs2 ) )
       => ( ( size_size_list_real @ ( slice_real @ From @ To @ Xs2 ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3287_slice__len,axiom,
    ! [From: nat,To: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs2 ) )
       => ( ( size_size_list_o @ ( slice_o @ From @ To @ Xs2 ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3288_slice__len,axiom,
    ! [From: nat,To: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs2 ) )
       => ( ( size_size_list_nat @ ( slice_nat @ From @ To @ Xs2 ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3289_slice__len,axiom,
    ! [From: nat,To: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs2 ) )
       => ( ( size_size_list_int @ ( slice_int @ From @ To @ Xs2 ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3290_double__diff,axiom,
    ! [A3: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_3291_double__diff,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( ord_less_eq_set_int @ B4 @ C2 )
       => ( ( minus_minus_set_int @ B4 @ ( minus_minus_set_int @ C2 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_3292_Diff__subset,axiom,
    ! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ A3 ) ).

% Diff_subset
thf(fact_3293_Diff__subset,axiom,
    ! [A3: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B4 ) @ A3 ) ).

% Diff_subset
thf(fact_3294_Diff__mono,axiom,
    ! [A3: set_nat,C2: set_nat,D4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ D4 @ B4 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3295_Diff__mono,axiom,
    ! [A3: set_int,C2: set_int,D4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ D4 @ B4 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B4 ) @ ( minus_minus_set_int @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3296_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X4 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X4 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3297_replicate__length__same,axiom,
    ! [Xs2: list_real,X4: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( X3 = X4 ) )
     => ( ( replicate_real @ ( size_size_list_real @ Xs2 ) @ X4 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3298_replicate__length__same,axiom,
    ! [Xs2: list_o,X4: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X4 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X4 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3299_replicate__length__same,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X4 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X4 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3300_replicate__length__same,axiom,
    ! [Xs2: list_int,X4: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( X3 = X4 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X4 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3301_replicate__eqI,axiom,
    ! [Xs2: list_set_nat,N: nat,X4: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs2 )
        = N )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_set_nat @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3302_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3303_replicate__eqI,axiom,
    ! [Xs2: list_real,N: nat,X4: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_real @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3304_replicate__eqI,axiom,
    ! [Xs2: list_o,N: nat,X4: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_o @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3305_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X4: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_nat @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3306_replicate__eqI,axiom,
    ! [Xs2: list_int,N: nat,X4: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X4 ) )
       => ( Xs2
          = ( replicate_int @ N @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_3307_sorted__replicate,axiom,
    ! [N: nat,X4: $o] : ( sorted_wrt_o @ ord_less_eq_o @ ( replicate_o @ N @ X4 ) ) ).

% sorted_replicate
thf(fact_3308_sorted__replicate,axiom,
    ! [N: nat,X4: rat] : ( sorted_wrt_rat @ ord_less_eq_rat @ ( replicate_rat @ N @ X4 ) ) ).

% sorted_replicate
thf(fact_3309_sorted__replicate,axiom,
    ! [N: nat,X4: num] : ( sorted_wrt_num @ ord_less_eq_num @ ( replicate_num @ N @ X4 ) ) ).

% sorted_replicate
thf(fact_3310_sorted__replicate,axiom,
    ! [N: nat,X4: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X4 ) ) ).

% sorted_replicate
thf(fact_3311_sorted__replicate,axiom,
    ! [N: nat,X4: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( replicate_int @ N @ X4 ) ) ).

% sorted_replicate
thf(fact_3312_subset__minus__empty,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ( minus_minus_set_real @ A3 @ B4 )
        = bot_bot_set_real ) ) ).

% subset_minus_empty
thf(fact_3313_subset__minus__empty,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B4 )
     => ( ( minus_minus_set_o @ A3 @ B4 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_3314_subset__minus__empty,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( minus_minus_set_nat @ A3 @ B4 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_3315_subset__minus__empty,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( minus_minus_set_int @ A3 @ B4 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_3316_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat
        @ ^ [X: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_nat @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3317_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real
        @ ^ [X: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_real @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3318_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] :
      ( ( map_real_VEBT_VEBT
        @ ^ [X: real] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3319_map__replicate__const,axiom,
    ! [K: $o,Lst: list_real] :
      ( ( map_real_o
        @ ^ [X: real] : K
        @ Lst )
      = ( replicate_o @ ( size_size_list_real @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3320_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] :
      ( ( map_o_VEBT_VEBT
        @ ^ [X: $o] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3321_map__replicate__const,axiom,
    ! [K: $o,Lst: list_o] :
      ( ( map_o_o
        @ ^ [X: $o] : K
        @ Lst )
      = ( replicate_o @ ( size_size_list_o @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3322_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3323_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] :
      ( ( map_nat_VEBT_VEBT
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3324_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] :
      ( ( map_nat_o
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_o @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3325_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] :
      ( ( map_int_VEBT_VEBT
        @ ^ [X: int] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3326_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_int,S4: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S5: set_int] :
              ( ( minus_minus_set_int @ S5 @ S3 )
              = ( minus_minus_set_int @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_3327_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_complex,S4: set_complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S5: set_complex] :
              ( ( minus_811609699411566653omplex @ S5 @ S3 )
              = ( minus_811609699411566653omplex @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_3328_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_Code_integer,S4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S5: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S5 @ S3 )
              = ( minus_2355218937544613996nteger @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_3329_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_nat,S4: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S5: set_nat] :
              ( ( minus_minus_set_nat @ S5 @ S3 )
              = ( minus_minus_set_nat @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_3330_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_3331_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( 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_3332_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X4 ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_3333_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X4 ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_3334_length__product,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( size_s3932428310213730859l_real @ ( product_real_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_3335_length__product,axiom,
    ! [Xs2: list_real,Ys: list_o] :
      ( ( size_s987546567493390085real_o @ ( product_real_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3336_length__product,axiom,
    ! [Xs2: list_real,Ys: list_nat] :
      ( ( size_s1877336372972134351al_nat @ ( product_real_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3337_length__product,axiom,
    ! [Xs2: list_real,Ys: list_int] :
      ( ( size_s8610625264895183403al_int @ ( product_real_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3338_length__product,axiom,
    ! [Xs2: list_o,Ys: list_real] :
      ( ( size_s2624279037499656343o_real @ ( product_o_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_3339_length__product,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3340_length__product,axiom,
    ! [Xs2: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3341_length__product,axiom,
    ! [Xs2: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3342_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_real] :
      ( ( size_s7910714270633306959t_real @ ( product_nat_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_3343_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3344_product__code,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( produc3886929683002245970T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ Ys ) )
      = ( set_Pr9182192707038809660T_VEBT
        @ ( concat8823785421885603304T_VEBT
          @ ( map_VE6710161768887062461T_VEBT
            @ ^ [X: vEBT_VEBT] : ( map_VE1720758354293053111T_VEBT @ ( produc537772716801021591T_VEBT @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3345_product__code,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( produc8336039005996768526T_real @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_real2 @ Ys ) )
      = ( set_Pr1087130671499945274T_real
        @ ( concat8867029399158321294T_real
          @ ( map_VE6455195641817765263T_real
            @ ^ [X: vEBT_VEBT] : ( map_re8618229306769252225T_real @ ( produc8117437818029410057T_real @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3346_product__code,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( produc2426872939560822710VEBT_o @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_o2 @ Ys ) )
      = ( set_Pr7708085864119495200VEBT_o
        @ ( concat318803602224216012VEBT_o
          @ ( map_VE92017342910548129VEBT_o
            @ ^ [X: vEBT_VEBT] : ( map_o_6754667662019005495VEBT_o @ ( produc8721562602347293563VEBT_o @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3347_product__code,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( produc1441754205559175824T_VEBT @ ( set_nat2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ Ys ) )
      = ( set_Pr5984661752051438084T_VEBT
        @ ( concat3360658388503054168T_VEBT
          @ ( map_na6744855865280072703T_VEBT
            @ ^ [X: nat] : ( map_VE5619196432070217417T_VEBT @ ( produc599794634098209291T_VEBT @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3348_product__code,axiom,
    ! [Xs2: list_nat,Ys: list_real] :
      ( ( produc6734486746367016272t_real @ ( set_nat2 @ Xs2 ) @ ( set_real2 @ Ys ) )
      = ( set_Pr7149346036329476978t_real
        @ ( concat8049025717632320286t_real
          @ ( map_na1385415807590506957t_real
            @ ^ [X: nat] : ( map_re7303296592222247787t_real @ ( produc7837566107596912789t_real @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3349_product__code,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( produc9051730707245535732_nat_o @ ( set_nat2 @ Xs2 ) @ ( set_o2 @ Ys ) )
      = ( set_Pr1291962091234853352_nat_o
        @ ( concat8133911138323470652_nat_o
          @ ( map_na5144441477028270563_nat_o
            @ ^ [X: nat] : ( map_o_7046636366338887185_nat_o @ ( product_Pair_nat_o @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3350_product__code,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT] :
      ( ( produc7150050738623674420T_VEBT @ ( set_real2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ Ys ) )
      = ( set_Pr8897343066327330088T_VEBT
        @ ( concat7453869757130930300T_VEBT
          @ ( map_re4732419559776791935T_VEBT
            @ ^ [X: real] : ( map_VE1210512188453230445T_VEBT @ ( produc6931449550656315951T_VEBT @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3351_product__code,axiom,
    ! [Xs2: list_real,Ys: list_nat] :
      ( ( produc2078423282641139152al_nat @ ( set_real2 @ Xs2 ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr3174298344852596722al_nat
        @ ( concat4073978026155440030al_nat
          @ ( map_re2667248788615319281al_nat
            @ ^ [X: real] : ( map_na1309146153819616583al_nat @ ( produc3181502643871035669al_nat @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3352_product__code,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( produc2998788613821421740l_real @ ( set_real2 @ Xs2 ) @ ( set_real2 @ Ys ) )
      = ( set_Pr5999470521830281550l_real
        @ ( concat431446536988871418l_real
          @ ( map_re4578740227252052045l_real
            @ ^ [X: real] : ( map_re3533924840734919879l_real @ ( produc4511245868158468465l_real @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3353_product__code,axiom,
    ! [Xs2: list_real,Ys: list_o] :
      ( ( produc674561262062614936real_o @ ( set_real2 @ Xs2 ) @ ( set_o2 @ Ys ) )
      = ( set_Pr5196769464307566348real_o
        @ ( concat1986395790136964192real_o
          @ ( map_re2784217942994236259real_o
            @ ^ [X: real] : ( map_o_7555934935399812789real_o @ ( product_Pair_real_o @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_3354_prod__decode__aux_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_3355_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_3356_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_3357_singletonI,axiom,
    ! [A: vEBT_VEBT] : ( member_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singletonI
thf(fact_3358_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_3359_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_3360_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_3361_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_3362_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_3363_insert__subset,axiom,
    ! [X4: $o,A3: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X4 @ A3 ) @ B4 )
      = ( ( member_o @ X4 @ B4 )
        & ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3364_insert__subset,axiom,
    ! [X4: nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A3 ) @ B4 )
      = ( ( member_nat @ X4 @ B4 )
        & ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3365_insert__subset,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X4 @ A3 ) @ B4 )
      = ( ( member_VEBT_VEBT @ X4 @ B4 )
        & ( ord_le4337996190870823476T_VEBT @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3366_insert__subset,axiom,
    ! [X4: real,A3: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X4 @ A3 ) @ B4 )
      = ( ( member_real @ X4 @ B4 )
        & ( ord_less_eq_set_real @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3367_insert__subset,axiom,
    ! [X4: set_nat,A3: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ A3 ) @ B4 )
      = ( ( member_set_nat @ X4 @ B4 )
        & ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3368_insert__subset,axiom,
    ! [X4: int,A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X4 @ A3 ) @ B4 )
      = ( ( member_int @ X4 @ B4 )
        & ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ).

% insert_subset
thf(fact_3369_singleton__conv,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : ( X = A ) )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_3370_singleton__conv,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( X = A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_3371_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( X = A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_3372_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( X = A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_3373_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X: real] : ( X = A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_3374_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X: $o] : ( X = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_3375_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_3376_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X: int] : ( X = A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_3377_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_3378_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_3379_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_3380_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_3381_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_3382_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_3383_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_3384_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_3385_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ A3 )
        = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_3386_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A3: set_real,B: real] :
      ( ( ( insert_real @ A @ A3 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_3387_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A3: set_o,B: $o] :
      ( ( ( insert_o @ A @ A3 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_3388_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A3: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A3 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_3389_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A3: set_int,B: int] :
      ( ( ( insert_int @ A @ A3 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_3390_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_3391_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A3: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_3392_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A3: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_3393_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A3: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_3394_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A3: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_3395_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( insert_VEBT_VEBT @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_3396_insert__Diff__single,axiom,
    ! [A: real,A3: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A3 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_3397_insert__Diff__single,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_3398_insert__Diff__single,axiom,
    ! [A: int,A3: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_3399_insert__Diff__single,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_3400_set__replicate,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X4 ) )
        = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_3401_set__replicate,axiom,
    ! [N: nat,X4: real] :
      ( ( N != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N @ X4 ) )
        = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_3402_set__replicate,axiom,
    ! [N: nat,X4: $o] :
      ( ( N != zero_zero_nat )
     => ( ( set_o2 @ ( replicate_o @ N @ X4 ) )
        = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% set_replicate
thf(fact_3403_set__replicate,axiom,
    ! [N: nat,X4: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X4 ) )
        = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_3404_set__replicate,axiom,
    ! [N: nat,X4: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X4 ) )
        = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_3405_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A5 )
              & ~ ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3406_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A5 )
              & ~ ( member_real @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3407_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A5 )
              & ~ ( member_complex @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3408_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A5 )
              & ~ ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3409_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A5 )
              & ~ ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3410_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A5 )
              & ~ ( member_int @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3411_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ~ ( member_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3412_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A5 )
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3413_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A5 )
            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3414_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X: complex] : ( member_complex @ X @ A5 )
            @ ^ [X: complex] : ( member_complex @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3415_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3416_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3417_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A5 )
            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3418_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 )
            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_3419_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3420_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3421_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3422_insert__Collect,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( insert_complex @ A @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3423_insert__Collect,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3424_insert__Collect,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3425_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3426_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_3427_insert__compr,axiom,
    ( insert_o
    = ( ^ [A2: $o,B5: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A2 )
              | ( member_o @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3428_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A2: vEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( X = A2 )
              | ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3429_insert__compr,axiom,
    ( insert_real
    = ( ^ [A2: real,B5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A2 )
              | ( member_real @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3430_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A2: complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( X = A2 )
              | ( member_complex @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3431_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A2: list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( X = A2 )
              | ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3432_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A2: set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( X = A2 )
              | ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3433_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A2: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A2 )
              | ( member_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3434_insert__compr,axiom,
    ( insert_int
    = ( ^ [A2: int,B5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( X = A2 )
              | ( member_int @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_3435_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_3436_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_3437_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_3438_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_3439_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_3440_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_3441_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_3442_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_3443_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_3444_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_3445_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_3446_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_3447_doubleton__eq__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,C: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ C @ ( insert_VEBT_VEBT @ D @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_3448_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_3449_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_3450_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_3451_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_3452_insert__not__empty,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ A3 )
     != bot_bo8194388402131092736T_VEBT ) ).

% insert_not_empty
thf(fact_3453_insert__not__empty,axiom,
    ! [A: real,A3: set_real] :
      ( ( insert_real @ A @ A3 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_3454_insert__not__empty,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ A3 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_3455_insert__not__empty,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat @ A @ A3 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_3456_insert__not__empty,axiom,
    ! [A: int,A3: set_int] :
      ( ( insert_int @ A @ A3 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_3457_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_3458_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_3459_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_3460_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_3461_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_3462_set__minus__singleton__eq,axiom,
    ! [X4: vEBT_VEBT,X7: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X4 @ X7 )
     => ( ( minus_5127226145743854075T_VEBT @ X7 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3463_set__minus__singleton__eq,axiom,
    ! [X4: set_nat,X7: set_set_nat] :
      ( ~ ( member_set_nat @ X4 @ X7 )
     => ( ( minus_2163939370556025621et_nat @ X7 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3464_set__minus__singleton__eq,axiom,
    ! [X4: real,X7: set_real] :
      ( ~ ( member_real @ X4 @ X7 )
     => ( ( minus_minus_set_real @ X7 @ ( insert_real @ X4 @ bot_bot_set_real ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3465_set__minus__singleton__eq,axiom,
    ! [X4: $o,X7: set_o] :
      ( ~ ( member_o @ X4 @ X7 )
     => ( ( minus_minus_set_o @ X7 @ ( insert_o @ X4 @ bot_bot_set_o ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3466_set__minus__singleton__eq,axiom,
    ! [X4: int,X7: set_int] :
      ( ~ ( member_int @ X4 @ X7 )
     => ( ( minus_minus_set_int @ X7 @ ( insert_int @ X4 @ bot_bot_set_int ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3467_set__minus__singleton__eq,axiom,
    ! [X4: nat,X7: set_nat] :
      ( ~ ( member_nat @ X4 @ X7 )
     => ( ( minus_minus_set_nat @ X7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_3468_Diff__insert__absorb,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X4 @ A3 ) @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3469_Diff__insert__absorb,axiom,
    ! [X4: set_nat,A3: set_set_nat] :
      ( ~ ( member_set_nat @ X4 @ A3 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A3 ) @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3470_Diff__insert__absorb,axiom,
    ! [X4: real,A3: set_real] :
      ( ~ ( member_real @ X4 @ A3 )
     => ( ( minus_minus_set_real @ ( insert_real @ X4 @ A3 ) @ ( insert_real @ X4 @ bot_bot_set_real ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3471_Diff__insert__absorb,axiom,
    ! [X4: $o,A3: set_o] :
      ( ~ ( member_o @ X4 @ A3 )
     => ( ( minus_minus_set_o @ ( insert_o @ X4 @ A3 ) @ ( insert_o @ X4 @ bot_bot_set_o ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3472_Diff__insert__absorb,axiom,
    ! [X4: int,A3: set_int] :
      ( ~ ( member_int @ X4 @ A3 )
     => ( ( minus_minus_set_int @ ( insert_int @ X4 @ A3 ) @ ( insert_int @ X4 @ bot_bot_set_int ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3473_Diff__insert__absorb,axiom,
    ! [X4: nat,A3: set_nat] :
      ( ~ ( member_nat @ X4 @ A3 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A3 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_3474_insert__minus__eq,axiom,
    ! [X4: vEBT_VEBT,Y: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( X4 != Y )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X4 @ A3 ) @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ X4 @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% insert_minus_eq
thf(fact_3475_insert__minus__eq,axiom,
    ! [X4: real,Y: real,A3: set_real] :
      ( ( X4 != Y )
     => ( ( minus_minus_set_real @ ( insert_real @ X4 @ A3 ) @ ( insert_real @ Y @ bot_bot_set_real ) )
        = ( insert_real @ X4 @ ( minus_minus_set_real @ A3 @ ( insert_real @ Y @ bot_bot_set_real ) ) ) ) ) ).

% insert_minus_eq
thf(fact_3476_insert__minus__eq,axiom,
    ! [X4: $o,Y: $o,A3: set_o] :
      ( ( X4 != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X4 @ A3 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X4 @ ( minus_minus_set_o @ A3 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_3477_insert__minus__eq,axiom,
    ! [X4: int,Y: int,A3: set_int] :
      ( ( X4 != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X4 @ A3 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X4 @ ( minus_minus_set_int @ A3 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_3478_insert__minus__eq,axiom,
    ! [X4: nat,Y: nat,A3: set_nat] :
      ( ( X4 != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A3 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X4 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_3479_Diff__insert2,axiom,
    ! [A3: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_3480_Diff__insert2,axiom,
    ! [A3: set_real,A: real,B4: set_real] :
      ( ( minus_minus_set_real @ A3 @ ( insert_real @ A @ B4 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_3481_Diff__insert2,axiom,
    ! [A3: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_3482_Diff__insert2,axiom,
    ! [A3: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A3 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_3483_Diff__insert2,axiom,
    ! [A3: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_3484_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3485_insert__Diff,axiom,
    ! [A: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A @ A3 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3486_insert__Diff,axiom,
    ! [A: real,A3: set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A3 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3487_insert__Diff,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3488_insert__Diff,axiom,
    ! [A: int,A3: set_int] :
      ( ( member_int @ A @ A3 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3489_insert__Diff,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_3490_Diff__insert,axiom,
    ! [A3: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ A @ B4 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ B4 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Diff_insert
thf(fact_3491_Diff__insert,axiom,
    ! [A3: set_real,A: real,B4: set_real] :
      ( ( minus_minus_set_real @ A3 @ ( insert_real @ A @ B4 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ B4 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_3492_Diff__insert,axiom,
    ! [A3: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_3493_Diff__insert,axiom,
    ! [A3: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A3 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ B4 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_3494_Diff__insert,axiom,
    ! [A3: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_3495_insert__subsetI,axiom,
    ! [X4: $o,A3: set_o,X7: set_o] :
      ( ( member_o @ X4 @ A3 )
     => ( ( ord_less_eq_set_o @ X7 @ A3 )
       => ( ord_less_eq_set_o @ ( insert_o @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3496_insert__subsetI,axiom,
    ! [X4: nat,A3: set_nat,X7: set_nat] :
      ( ( member_nat @ X4 @ A3 )
     => ( ( ord_less_eq_set_nat @ X7 @ A3 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3497_insert__subsetI,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,X7: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( ord_le4337996190870823476T_VEBT @ X7 @ A3 )
       => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3498_insert__subsetI,axiom,
    ! [X4: real,A3: set_real,X7: set_real] :
      ( ( member_real @ X4 @ A3 )
     => ( ( ord_less_eq_set_real @ X7 @ A3 )
       => ( ord_less_eq_set_real @ ( insert_real @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3499_insert__subsetI,axiom,
    ! [X4: set_nat,A3: set_set_nat,X7: set_set_nat] :
      ( ( member_set_nat @ X4 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ X7 @ A3 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3500_insert__subsetI,axiom,
    ! [X4: int,A3: set_int,X7: set_int] :
      ( ( member_int @ X4 @ A3 )
     => ( ( ord_less_eq_set_int @ X7 @ A3 )
       => ( ord_less_eq_set_int @ ( insert_int @ X4 @ X7 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_3501_insert__mono,axiom,
    ! [C2: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_3502_insert__mono,axiom,
    ! [C2: set_VEBT_VEBT,D4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ C2 @ D4 )
     => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C2 ) @ ( insert_VEBT_VEBT @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_3503_insert__mono,axiom,
    ! [C2: set_o,D4: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_3504_insert__mono,axiom,
    ! [C2: set_real,D4: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C2 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C2 ) @ ( insert_real @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_3505_insert__mono,axiom,
    ! [C2: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C2 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_3506_subset__insert,axiom,
    ! [X4: $o,A3: set_o,B4: set_o] :
      ( ~ ( member_o @ X4 @ A3 )
     => ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X4 @ B4 ) )
        = ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3507_subset__insert,axiom,
    ! [X4: nat,A3: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X4 @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X4 @ B4 ) )
        = ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3508_subset__insert,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ B4 ) )
        = ( ord_le4337996190870823476T_VEBT @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3509_subset__insert,axiom,
    ! [X4: real,A3: set_real,B4: set_real] :
      ( ~ ( member_real @ X4 @ A3 )
     => ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X4 @ B4 ) )
        = ( ord_less_eq_set_real @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3510_subset__insert,axiom,
    ! [X4: set_nat,A3: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X4 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X4 @ B4 ) )
        = ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3511_subset__insert,axiom,
    ! [X4: int,A3: set_int,B4: set_int] :
      ( ~ ( member_int @ X4 @ A3 )
     => ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X4 @ B4 ) )
        = ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ).

% subset_insert
thf(fact_3512_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_3513_subset__insertI,axiom,
    ! [B4: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B4 @ ( insert_VEBT_VEBT @ A @ B4 ) ) ).

% subset_insertI
thf(fact_3514_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).

% subset_insertI
thf(fact_3515_subset__insertI,axiom,
    ! [B4: set_real,A: real] : ( ord_less_eq_set_real @ B4 @ ( insert_real @ A @ B4 ) ) ).

% subset_insertI
thf(fact_3516_subset__insertI,axiom,
    ! [B4: set_int,A: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A @ B4 ) ) ).

% subset_insertI
thf(fact_3517_subset__insertI2,axiom,
    ! [A3: set_nat,B4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_3518_subset__insertI2,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_3519_subset__insertI2,axiom,
    ! [A3: set_o,B4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B4 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_3520_subset__insertI2,axiom,
    ! [A3: set_real,B4: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ord_less_eq_set_real @ A3 @ ( insert_real @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_3521_subset__insertI2,axiom,
    ! [A3: set_int,B4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ A3 @ ( insert_int @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_3522_subset__Diff__insert,axiom,
    ! [A3: set_o,B4: set_o,X4: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X4 @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B4 @ C2 ) )
        & ~ ( member_o @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3523_subset__Diff__insert,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,X4: vEBT_VEBT,C2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ B4 @ ( insert_VEBT_VEBT @ X4 @ C2 ) ) )
      = ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ B4 @ C2 ) )
        & ~ ( member_VEBT_VEBT @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3524_subset__Diff__insert,axiom,
    ! [A3: set_real,B4: set_real,X4: real,C2: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B4 @ ( insert_real @ X4 @ C2 ) ) )
      = ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B4 @ C2 ) )
        & ~ ( member_real @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3525_subset__Diff__insert,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,X4: set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B4 @ ( insert_set_nat @ X4 @ C2 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B4 @ C2 ) )
        & ~ ( member_set_nat @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3526_subset__Diff__insert,axiom,
    ! [A3: set_nat,B4: set_nat,X4: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X4 @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ C2 ) )
        & ~ ( member_nat @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3527_subset__Diff__insert,axiom,
    ! [A3: set_int,B4: set_int,X4: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B4 @ ( insert_int @ X4 @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B4 @ C2 ) )
        & ~ ( member_int @ X4 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_3528_Collect__conv__if,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_3529_Collect__conv__if,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_3530_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_3531_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_3532_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_3533_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_3534_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_3535_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_3536_Collect__conv__if2,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_3537_Collect__conv__if2,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_3538_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_3539_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_3540_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_3541_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_3542_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_3543_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_3544_finite_Ocases,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A7: set_VEBT_VEBT] :
              ( ? [A4: vEBT_VEBT] :
                  ( A
                  = ( insert_VEBT_VEBT @ A4 @ A7 ) )
             => ~ ( finite5795047828879050333T_VEBT @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3545_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A4: complex] :
                  ( A
                  = ( insert_complex @ A4 @ A7 ) )
             => ~ ( finite3207457112153483333omplex @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3546_finite_Ocases,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ~ ! [A7: set_Code_integer] :
              ( ? [A4: code_integer] :
                  ( A
                  = ( insert_Code_integer @ A4 @ A7 ) )
             => ~ ( finite6017078050557962740nteger @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3547_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A7: set_real] :
              ( ? [A4: real] :
                  ( A
                  = ( insert_real @ A4 @ A7 ) )
             => ~ ( finite_finite_real @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3548_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A4: $o] :
                  ( A
                  = ( insert_o @ A4 @ A7 ) )
             => ~ ( finite_finite_o @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3549_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A4: nat] :
                  ( A
                  = ( insert_nat @ A4 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3550_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A4: int] :
                  ( A
                  = ( insert_int @ A4 @ A7 ) )
             => ~ ( finite_finite_int @ A7 ) ) ) ) ).

% finite.cases
thf(fact_3551_finite_Osimps,axiom,
    ( finite5795047828879050333T_VEBT
    = ( ^ [A2: set_VEBT_VEBT] :
          ( ( A2 = bot_bo8194388402131092736T_VEBT )
          | ? [A5: set_VEBT_VEBT,B2: vEBT_VEBT] :
              ( ( A2
                = ( insert_VEBT_VEBT @ B2 @ A5 ) )
              & ( finite5795047828879050333T_VEBT @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3552_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A2: set_complex] :
          ( ( A2 = bot_bot_set_complex )
          | ? [A5: set_complex,B2: complex] :
              ( ( A2
                = ( insert_complex @ B2 @ A5 ) )
              & ( finite3207457112153483333omplex @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3553_finite_Osimps,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A2: set_Code_integer] :
          ( ( A2 = bot_bo3990330152332043303nteger )
          | ? [A5: set_Code_integer,B2: code_integer] :
              ( ( A2
                = ( insert_Code_integer @ B2 @ A5 ) )
              & ( finite6017078050557962740nteger @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3554_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A2: set_real] :
          ( ( A2 = bot_bot_set_real )
          | ? [A5: set_real,B2: real] :
              ( ( A2
                = ( insert_real @ B2 @ A5 ) )
              & ( finite_finite_real @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3555_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A2: set_o] :
          ( ( A2 = bot_bot_set_o )
          | ? [A5: set_o,B2: $o] :
              ( ( A2
                = ( insert_o @ B2 @ A5 ) )
              & ( finite_finite_o @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3556_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A2: set_nat] :
          ( ( A2 = bot_bot_set_nat )
          | ? [A5: set_nat,B2: nat] :
              ( ( A2
                = ( insert_nat @ B2 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3557_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A2: set_int] :
          ( ( A2 = bot_bot_set_int )
          | ? [A5: set_int,B2: int] :
              ( ( A2
                = ( insert_int @ B2 @ A5 ) )
              & ( finite_finite_int @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_3558_finite__induct,axiom,
    ! [F5: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F6 )
             => ( ~ ( member_VEBT_VEBT @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3559_finite__induct,axiom,
    ! [F5: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X3: set_nat,F6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F6 )
             => ( ~ ( member_set_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_set_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3560_finite__induct,axiom,
    ! [F5: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,F6: set_complex] :
              ( ( finite3207457112153483333omplex @ F6 )
             => ( ~ ( member_complex @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_complex @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3561_finite__induct,axiom,
    ! [F5: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,F6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F6 )
             => ( ~ ( member_Code_integer @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_Code_integer @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3562_finite__induct,axiom,
    ! [F5: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,F6: set_real] :
              ( ( finite_finite_real @ F6 )
             => ( ~ ( member_real @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_real @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3563_finite__induct,axiom,
    ! [F5: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,F6: set_o] :
              ( ( finite_finite_o @ F6 )
             => ( ~ ( member_o @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_o @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3564_finite__induct,axiom,
    ! [F5: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3565_finite__induct,axiom,
    ! [F5: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,F6: set_int] :
              ( ( finite_finite_int @ F6 )
             => ( ~ ( member_int @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_int @ X3 @ F6 ) ) ) ) )
         => ( P @ F5 ) ) ) ) ).

% finite_induct
thf(fact_3566_finite__ne__induct,axiom,
    ! [F5: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( F5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] : ( P @ ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) )
         => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( F6 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ ( member_VEBT_VEBT @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_VEBT_VEBT @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3567_finite__ne__induct,axiom,
    ! [F5: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( F5 != bot_bot_set_set_nat )
       => ( ! [X3: set_nat] : ( P @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
         => ( ! [X3: set_nat,F6: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F6 )
               => ( ( F6 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_set_nat @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3568_finite__ne__induct,axiom,
    ! [F5: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( F5 != bot_bot_set_complex )
       => ( ! [X3: complex] : ( P @ ( insert_complex @ X3 @ bot_bot_set_complex ) )
         => ( ! [X3: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( F6 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_complex @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3569_finite__ne__induct,axiom,
    ! [F5: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( F5 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] : ( P @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) )
         => ( ! [X3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( F6 != bot_bo3990330152332043303nteger )
                 => ( ~ ( member_Code_integer @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_Code_integer @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3570_finite__ne__induct,axiom,
    ! [F5: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( F5 != bot_bot_set_real )
       => ( ! [X3: real] : ( P @ ( insert_real @ X3 @ bot_bot_set_real ) )
         => ( ! [X3: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( F6 != bot_bot_set_real )
                 => ( ~ ( member_real @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_real @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3571_finite__ne__induct,axiom,
    ! [F5: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( F5 != bot_bot_set_o )
       => ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
         => ( ! [X3: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( F6 != bot_bot_set_o )
                 => ( ~ ( member_o @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_o @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3572_finite__ne__induct,axiom,
    ! [F5: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( F5 != bot_bot_set_nat )
       => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
         => ( ! [X3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( F6 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3573_finite__ne__induct,axiom,
    ! [F5: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( F5 != bot_bot_set_int )
       => ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
         => ( ! [X3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( F6 != bot_bot_set_int )
                 => ( ~ ( member_int @ X3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_int @ X3 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_3574_infinite__finite__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,A3: set_VEBT_VEBT] :
      ( ! [A7: set_VEBT_VEBT] :
          ( ~ ( finite5795047828879050333T_VEBT @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F6 )
             => ( ~ ( member_VEBT_VEBT @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3575_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A3: set_set_nat] :
      ( ! [A7: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X3: set_nat,F6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F6 )
             => ( ~ ( member_set_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_set_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3576_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A3: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,F6: set_complex] :
              ( ( finite3207457112153483333omplex @ F6 )
             => ( ~ ( member_complex @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_complex @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3577_infinite__finite__induct,axiom,
    ! [P: set_Code_integer > $o,A3: set_Code_integer] :
      ( ! [A7: set_Code_integer] :
          ( ~ ( finite6017078050557962740nteger @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,F6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F6 )
             => ( ~ ( member_Code_integer @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_Code_integer @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3578_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A3: set_real] :
      ( ! [A7: set_real] :
          ( ~ ( finite_finite_real @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,F6: set_real] :
              ( ( finite_finite_real @ F6 )
             => ( ~ ( member_real @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_real @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3579_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A3: set_o] :
      ( ! [A7: set_o] :
          ( ~ ( finite_finite_o @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,F6: set_o] :
              ( ( finite_finite_o @ F6 )
             => ( ~ ( member_o @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_o @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3580_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A3: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3581_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A3: set_int] :
      ( ! [A7: set_int] :
          ( ~ ( finite_finite_int @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,F6: set_int] :
              ( ( finite_finite_int @ F6 )
             => ( ~ ( member_int @ X3 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_int @ X3 @ F6 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_3582_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_3583_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_3584_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_3585_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ ( finite_finite_real @ S3 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_3586_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_3587_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_3588_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_3589_infinite__coinduct,axiom,
    ! [X7: set_VEBT_VEBT > $o,A3: set_VEBT_VEBT] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_VEBT_VEBT] :
            ( ( X7 @ A7 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A7 )
                & ( ( X7 @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) )
                  | ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
       => ~ ( finite5795047828879050333T_VEBT @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3590_infinite__coinduct,axiom,
    ! [X7: set_complex > $o,A3: set_complex] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_complex] :
            ( ( X7 @ A7 )
           => ? [X6: complex] :
                ( ( member_complex @ X6 @ A7 )
                & ( ( X7 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3591_infinite__coinduct,axiom,
    ! [X7: set_Code_integer > $o,A3: set_Code_integer] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_Code_integer] :
            ( ( X7 @ A7 )
           => ? [X6: code_integer] :
                ( ( member_Code_integer @ X6 @ A7 )
                & ( ( X7 @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) )
                  | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
       => ~ ( finite6017078050557962740nteger @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3592_infinite__coinduct,axiom,
    ! [X7: set_real > $o,A3: set_real] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_real] :
            ( ( X7 @ A7 )
           => ? [X6: real] :
                ( ( member_real @ X6 @ A7 )
                & ( ( X7 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X6 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3593_infinite__coinduct,axiom,
    ! [X7: set_o > $o,A3: set_o] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_o] :
            ( ( X7 @ A7 )
           => ? [X6: $o] :
                ( ( member_o @ X6 @ A7 )
                & ( ( X7 @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3594_infinite__coinduct,axiom,
    ! [X7: set_int > $o,A3: set_int] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_int] :
            ( ( X7 @ A7 )
           => ? [X6: int] :
                ( ( member_int @ X6 @ A7 )
                & ( ( X7 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3595_infinite__coinduct,axiom,
    ! [X7: set_nat > $o,A3: set_nat] :
      ( ( X7 @ A3 )
     => ( ! [A7: set_nat] :
            ( ( X7 @ A7 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A7 )
                & ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3596_finite__empty__induct,axiom,
    ! [A3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: vEBT_VEBT,A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( member_VEBT_VEBT @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ A4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
         => ( P @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% finite_empty_induct
thf(fact_3597_finite__empty__induct,axiom,
    ! [A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: set_nat,A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( member_set_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_3598_finite__empty__induct,axiom,
    ! [A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: complex,A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( member_complex @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_3599_finite__empty__induct,axiom,
    ! [A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( member_Code_integer @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ A4 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
         => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).

% finite_empty_induct
thf(fact_3600_finite__empty__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( member_real @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_3601_finite__empty__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( member_o @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_3602_finite__empty__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_3603_finite__empty__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_3604_subset__singleton__iff,axiom,
    ! [X7: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X7 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( X7 = bot_bo8194388402131092736T_VEBT )
        | ( X7
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singleton_iff
thf(fact_3605_subset__singleton__iff,axiom,
    ! [X7: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X7 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X7 = bot_bot_set_real )
        | ( X7
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_3606_subset__singleton__iff,axiom,
    ! [X7: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X7 = bot_bot_set_o )
        | ( X7
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_3607_subset__singleton__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X7 = bot_bot_set_nat )
        | ( X7
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_3608_subset__singleton__iff,axiom,
    ! [X7: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X7 = bot_bot_set_int )
        | ( X7
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_3609_subset__singletonD,axiom,
    ! [A3: set_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
     => ( ( A3 = bot_bo8194388402131092736T_VEBT )
        | ( A3
          = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singletonD
thf(fact_3610_subset__singletonD,axiom,
    ! [A3: set_real,X4: real] :
      ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) )
     => ( ( A3 = bot_bot_set_real )
        | ( A3
          = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_3611_subset__singletonD,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) )
     => ( ( A3 = bot_bot_set_o )
        | ( A3
          = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_3612_subset__singletonD,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
     => ( ( A3 = bot_bot_set_nat )
        | ( A3
          = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_3613_subset__singletonD,axiom,
    ! [A3: set_int,X4: int] :
      ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) )
     => ( ( A3 = bot_bot_set_int )
        | ( A3
          = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_3614_Diff__single__insert,axiom,
    ! [A3: set_VEBT_VEBT,X4: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) @ B4 )
     => ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_3615_Diff__single__insert,axiom,
    ! [A3: set_real,X4: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ B4 )
     => ( ord_less_eq_set_real @ A3 @ ( insert_real @ X4 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_3616_Diff__single__insert,axiom,
    ! [A3: set_o,X4: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B4 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ X4 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_3617_Diff__single__insert,axiom,
    ! [A3: set_nat,X4: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B4 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X4 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_3618_Diff__single__insert,axiom,
    ! [A3: set_int,X4: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ B4 )
     => ( ord_less_eq_set_int @ A3 @ ( insert_int @ X4 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_3619_subset__insert__iff,axiom,
    ! [A3: set_VEBT_VEBT,X4: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ B4 ) )
      = ( ( ( member_VEBT_VEBT @ X4 @ A3 )
         => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) )
        & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
         => ( ord_le4337996190870823476T_VEBT @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3620_subset__insert__iff,axiom,
    ! [A3: set_set_nat,X4: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X4 @ B4 ) )
      = ( ( ( member_set_nat @ X4 @ A3 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B4 ) )
        & ( ~ ( member_set_nat @ X4 @ A3 )
         => ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3621_subset__insert__iff,axiom,
    ! [A3: set_real,X4: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X4 @ B4 ) )
      = ( ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ B4 ) )
        & ( ~ ( member_real @ X4 @ A3 )
         => ( ord_less_eq_set_real @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3622_subset__insert__iff,axiom,
    ! [A3: set_o,X4: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X4 @ B4 ) )
      = ( ( ( member_o @ X4 @ A3 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B4 ) )
        & ( ~ ( member_o @ X4 @ A3 )
         => ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3623_subset__insert__iff,axiom,
    ! [A3: set_nat,X4: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X4 @ B4 ) )
      = ( ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3624_subset__insert__iff,axiom,
    ! [A3: set_int,X4: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X4 @ B4 ) )
      = ( ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ B4 ) )
        & ( ~ ( member_int @ X4 @ A3 )
         => ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_3625_remove__subset,axiom,
    ! [X4: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ S3 )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3626_remove__subset,axiom,
    ! [X4: set_nat,S3: set_set_nat] :
      ( ( member_set_nat @ X4 @ S3 )
     => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ S3 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3627_remove__subset,axiom,
    ! [X4: real,S3: set_real] :
      ( ( member_real @ X4 @ S3 )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3628_remove__subset,axiom,
    ! [X4: $o,S3: set_o] :
      ( ( member_o @ X4 @ S3 )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3629_remove__subset,axiom,
    ! [X4: int,S3: set_int] :
      ( ( member_int @ X4 @ S3 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3630_remove__subset,axiom,
    ! [X4: nat,S3: set_nat] :
      ( ( member_nat @ X4 @ S3 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ S3 ) ) ).

% remove_subset
thf(fact_3631_product__concat__map,axiom,
    ( product_nat_nat
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Ys3 )
            @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_3632_product__concat__map,axiom,
    ( produc7295137177222721919BT_nat
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_nat] :
          ( concat4407583305730369330BT_nat
          @ ( map_VE4456067000678548787BT_nat
            @ ^ [X: vEBT_VEBT] : ( map_na4631810538828370761BT_nat @ ( produc738532404422230701BT_nat @ X ) @ Ys3 )
            @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_3633_product__concat__map,axiom,
    ( product_int_int
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( concat4512918505337516154nt_int
          @ ( map_in7266296235447420877nt_int
            @ ^ [X: int] : ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ X ) @ Ys3 )
            @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_3634_product__concat__map,axiom,
    ( produc8792966785426426881nteger
    = ( ^ [Xs: list_Code_integer,Ys3: list_Code_integer] :
          ( concat6978052072357435484nteger
          @ ( map_Co2570546026900698046nteger
            @ ^ [X: code_integer] : ( map_Co3589949550033412536nteger @ ( produc1086072967326762835nteger @ X ) @ Ys3 )
            @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_3635_product__concat__map,axiom,
    ( product_nat_num
    = ( ^ [Xs: list_nat,Ys3: list_num] :
          ( concat8399659749894496012at_num
          @ ( map_na277125420730788027at_num
            @ ^ [X: nat] : ( map_nu4721551698833171051at_num @ ( product_Pair_nat_num @ X ) @ Ys3 )
            @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_3636_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3637_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3638_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3639_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_real @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3640_finite__ranking__induct,axiom,
    ! [S3: set_o,P: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,S6: set_o] :
              ( ( finite_finite_o @ S6 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_o @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3641_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3642_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S6: set_int] :
              ( ( finite_finite_int @ S6 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_int @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3643_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y5 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3644_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3645_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_3646_finite__linorder__max__induct,axiom,
    ! [A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A7 )
                   => ( ord_le6747313008572928689nteger @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3647_finite__linorder__max__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A7 )
                   => ( ord_less_o @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3648_finite__linorder__max__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X6: real] :
                    ( ( member_real @ X6 @ A7 )
                   => ( ord_less_real @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3649_finite__linorder__max__induct,axiom,
    ! [A3: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A3 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A7 )
                   => ( ord_less_rat @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3650_finite__linorder__max__induct,axiom,
    ! [A3: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A3 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A7 )
                   => ( ord_less_num @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3651_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3652_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A7 )
                   => ( ord_less_int @ X6 @ B3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3653_finite__linorder__min__induct,axiom,
    ! [A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A7 )
                   => ( ord_le6747313008572928689nteger @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3654_finite__linorder__min__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A7 )
                   => ( ord_less_o @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3655_finite__linorder__min__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X6: real] :
                    ( ( member_real @ X6 @ A7 )
                   => ( ord_less_real @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3656_finite__linorder__min__induct,axiom,
    ! [A3: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A3 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A7 )
                   => ( ord_less_rat @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3657_finite__linorder__min__induct,axiom,
    ! [A3: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A3 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A7 )
                   => ( ord_less_num @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3658_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3659_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A7 )
                   => ( ord_less_int @ B3 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B3 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3660_finite__subset__induct,axiom,
    ! [F5: set_VEBT_VEBT,A3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT @ F5 @ A3 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A4: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A4 @ A3 )
                 => ( ~ ( member_VEBT_VEBT @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_VEBT_VEBT @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3661_finite__subset__induct,axiom,
    ! [F5: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( ord_le6893508408891458716et_nat @ F5 @ A3 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F6: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F6 )
               => ( ( member_set_nat @ A4 @ A3 )
                 => ( ~ ( member_set_nat @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_set_nat @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3662_finite__subset__induct,axiom,
    ! [F5: set_complex,A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex @ F5 @ A3 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A4: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A4 @ A3 )
                 => ( ~ ( member_complex @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_complex @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3663_finite__subset__induct,axiom,
    ! [F5: set_Code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A3 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A4: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A4 @ A3 )
                 => ( ~ ( member_Code_integer @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_Code_integer @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3664_finite__subset__induct,axiom,
    ! [F5: set_real,A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real @ F5 @ A3 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A4: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A4 @ A3 )
                 => ( ~ ( member_real @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_real @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3665_finite__subset__induct,axiom,
    ! [F5: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o @ F5 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A4: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A4 @ A3 )
                 => ( ~ ( member_o @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_o @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3666_finite__subset__induct,axiom,
    ! [F5: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A4 @ A3 )
                 => ( ~ ( member_nat @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3667_finite__subset__induct,axiom,
    ! [F5: set_int,A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A3 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A4: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A4 @ A3 )
                 => ( ~ ( member_int @ A4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_int @ A4 @ F6 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3668_finite__subset__induct_H,axiom,
    ! [F5: set_VEBT_VEBT,A3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT @ F5 @ A3 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A4: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A4 @ A3 )
                 => ( ( ord_le4337996190870823476T_VEBT @ F6 @ A3 )
                   => ( ~ ( member_VEBT_VEBT @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_VEBT_VEBT @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3669_finite__subset__induct_H,axiom,
    ! [F5: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( ord_le6893508408891458716et_nat @ F5 @ A3 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F6: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F6 )
               => ( ( member_set_nat @ A4 @ A3 )
                 => ( ( ord_le6893508408891458716et_nat @ F6 @ A3 )
                   => ( ~ ( member_set_nat @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_set_nat @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3670_finite__subset__induct_H,axiom,
    ! [F5: set_complex,A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex @ F5 @ A3 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A4: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A4 @ A3 )
                 => ( ( ord_le211207098394363844omplex @ F6 @ A3 )
                   => ( ~ ( member_complex @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_complex @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3671_finite__subset__induct_H,axiom,
    ! [F5: set_Code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A3 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A4: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A4 @ A3 )
                 => ( ( ord_le7084787975880047091nteger @ F6 @ A3 )
                   => ( ~ ( member_Code_integer @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_Code_integer @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3672_finite__subset__induct_H,axiom,
    ! [F5: set_real,A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real @ F5 @ A3 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A4: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A4 @ A3 )
                 => ( ( ord_less_eq_set_real @ F6 @ A3 )
                   => ( ~ ( member_real @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_real @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3673_finite__subset__induct_H,axiom,
    ! [F5: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o @ F5 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A4: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A4 @ A3 )
                 => ( ( ord_less_eq_set_o @ F6 @ A3 )
                   => ( ~ ( member_o @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_o @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3674_finite__subset__induct_H,axiom,
    ! [F5: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A4 @ A3 )
                 => ( ( ord_less_eq_set_nat @ F6 @ A3 )
                   => ( ~ ( member_nat @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_nat @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3675_finite__subset__induct_H,axiom,
    ! [F5: set_int,A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A3 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A4: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A4 @ A3 )
                 => ( ( ord_less_eq_set_int @ F6 @ A3 )
                   => ( ~ ( member_int @ A4 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_int @ A4 @ F6 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3676_remove__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,B4: set_VEBT_VEBT] :
      ( ( P @ bot_bo8194388402131092736T_VEBT )
     => ( ( ~ ( finite5795047828879050333T_VEBT @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A7 @ B4 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ A7 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3677_remove__induct,axiom,
    ! [P: set_set_nat > $o,B4: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B4 )
                 => ( ! [X6: set_nat] :
                        ( ( member_set_nat @ X6 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3678_remove__induct,axiom,
    ! [P: set_complex > $o,B4: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B4 )
                 => ( ! [X6: complex] :
                        ( ( member_complex @ X6 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3679_remove__induct,axiom,
    ! [P: set_Code_integer > $o,B4: set_Code_integer] :
      ( ( P @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A7 @ B4 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A7 )
                       => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3680_remove__induct,axiom,
    ! [P: set_real > $o,B4: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B4 )
                 => ( ! [X6: real] :
                        ( ( member_real @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3681_remove__induct,axiom,
    ! [P: set_o > $o,B4: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B4 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3682_remove__induct,axiom,
    ! [P: set_nat > $o,B4: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B4 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3683_remove__induct,axiom,
    ! [P: set_int > $o,B4: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B4 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_3684_finite__remove__induct,axiom,
    ! [B4: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B4 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A7 @ B4 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ A7 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X6 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3685_finite__remove__induct,axiom,
    ! [B4: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B4 )
                 => ( ! [X6: set_nat] :
                        ( ( member_set_nat @ X6 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3686_finite__remove__induct,axiom,
    ! [B4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B4 )
                 => ( ! [X6: complex] :
                        ( ( member_complex @ X6 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X6 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3687_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A7 @ B4 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A7 )
                       => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3688_finite__remove__induct,axiom,
    ! [B4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B4 )
                 => ( ! [X6: real] :
                        ( ( member_real @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X6 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3689_finite__remove__induct,axiom,
    ! [B4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B4 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3690_finite__remove__induct,axiom,
    ! [B4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B4 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3691_finite__remove__induct,axiom,
    ! [B4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B4 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_3692_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T5: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X6 @ ( minus_5127226145743854075T_VEBT @ S3 @ T5 ) )
                    & ( P @ ( insert_VEBT_VEBT @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3693_finite__induct__select,axiom,
    ! [S3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T5: set_complex] :
              ( ( ord_less_set_complex @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: complex] :
                    ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ S3 @ T5 ) )
                    & ( P @ ( insert_complex @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3694_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [T5: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S3 @ T5 ) )
                    & ( P @ ( insert_Code_integer @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3695_finite__induct__select,axiom,
    ! [S3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T5: set_real] :
              ( ( ord_less_set_real @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: real] :
                    ( ( member_real @ X6 @ ( minus_minus_set_real @ S3 @ T5 ) )
                    & ( P @ ( insert_real @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3696_finite__induct__select,axiom,
    ! [S3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T5: set_o] :
              ( ( ord_less_set_o @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: $o] :
                    ( ( member_o @ X6 @ ( minus_minus_set_o @ S3 @ T5 ) )
                    & ( P @ ( insert_o @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3697_finite__induct__select,axiom,
    ! [S3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T5: set_int] :
              ( ( ord_less_set_int @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: int] :
                    ( ( member_int @ X6 @ ( minus_minus_set_int @ S3 @ T5 ) )
                    & ( P @ ( insert_int @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3698_finite__induct__select,axiom,
    ! [S3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T5: set_nat] :
              ( ( ord_less_set_nat @ T5 @ S3 )
             => ( ( P @ T5 )
               => ? [X6: nat] :
                    ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S3 @ T5 ) )
                    & ( P @ ( insert_nat @ X6 @ T5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3699_psubset__insert__iff,axiom,
    ! [A3: set_VEBT_VEBT,X4: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ B4 ) )
      = ( ( ( member_VEBT_VEBT @ X4 @ B4 )
         => ( ord_le3480810397992357184T_VEBT @ A3 @ B4 ) )
        & ( ~ ( member_VEBT_VEBT @ X4 @ B4 )
         => ( ( ( member_VEBT_VEBT @ X4 @ A3 )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) )
            & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
             => ( ord_le4337996190870823476T_VEBT @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3700_psubset__insert__iff,axiom,
    ! [A3: set_set_nat,X4: set_nat,B4: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ ( insert_set_nat @ X4 @ B4 ) )
      = ( ( ( member_set_nat @ X4 @ B4 )
         => ( ord_less_set_set_nat @ A3 @ B4 ) )
        & ( ~ ( member_set_nat @ X4 @ B4 )
         => ( ( ( member_set_nat @ X4 @ A3 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B4 ) )
            & ( ~ ( member_set_nat @ X4 @ A3 )
             => ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3701_psubset__insert__iff,axiom,
    ! [A3: set_real,X4: real,B4: set_real] :
      ( ( ord_less_set_real @ A3 @ ( insert_real @ X4 @ B4 ) )
      = ( ( ( member_real @ X4 @ B4 )
         => ( ord_less_set_real @ A3 @ B4 ) )
        & ( ~ ( member_real @ X4 @ B4 )
         => ( ( ( member_real @ X4 @ A3 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ B4 ) )
            & ( ~ ( member_real @ X4 @ A3 )
             => ( ord_less_eq_set_real @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3702_psubset__insert__iff,axiom,
    ! [A3: set_o,X4: $o,B4: set_o] :
      ( ( ord_less_set_o @ A3 @ ( insert_o @ X4 @ B4 ) )
      = ( ( ( member_o @ X4 @ B4 )
         => ( ord_less_set_o @ A3 @ B4 ) )
        & ( ~ ( member_o @ X4 @ B4 )
         => ( ( ( member_o @ X4 @ A3 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B4 ) )
            & ( ~ ( member_o @ X4 @ A3 )
             => ( ord_less_eq_set_o @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3703_psubset__insert__iff,axiom,
    ! [A3: set_nat,X4: nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A3 @ ( insert_nat @ X4 @ B4 ) )
      = ( ( ( member_nat @ X4 @ B4 )
         => ( ord_less_set_nat @ A3 @ B4 ) )
        & ( ~ ( member_nat @ X4 @ B4 )
         => ( ( ( member_nat @ X4 @ A3 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B4 ) )
            & ( ~ ( member_nat @ X4 @ A3 )
             => ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3704_psubset__insert__iff,axiom,
    ! [A3: set_int,X4: int,B4: set_int] :
      ( ( ord_less_set_int @ A3 @ ( insert_int @ X4 @ B4 ) )
      = ( ( ( member_int @ X4 @ B4 )
         => ( ord_less_set_int @ A3 @ B4 ) )
        & ( ~ ( member_int @ X4 @ B4 )
         => ( ( ( member_int @ X4 @ A3 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ B4 ) )
            & ( ~ ( member_int @ X4 @ A3 )
             => ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_3705_set__replicate__Suc,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X4 ) )
      = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_3706_set__replicate__Suc,axiom,
    ! [N: nat,X4: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X4 ) )
      = ( insert_real @ X4 @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_3707_set__replicate__Suc,axiom,
    ! [N: nat,X4: $o] :
      ( ( set_o2 @ ( replicate_o @ ( suc @ N ) @ X4 ) )
      = ( insert_o @ X4 @ bot_bot_set_o ) ) ).

% set_replicate_Suc
thf(fact_3708_set__replicate__Suc,axiom,
    ! [N: nat,X4: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X4 ) )
      = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_3709_set__replicate__Suc,axiom,
    ! [N: nat,X4: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X4 ) )
      = ( insert_int @ X4 @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_3710_set__replicate__conv__if,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X4 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X4 ) )
          = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3711_set__replicate__conv__if,axiom,
    ! [N: nat,X4: real] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X4 ) )
          = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X4 ) )
          = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3712_set__replicate__conv__if,axiom,
    ! [N: nat,X4: $o] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X4 ) )
          = bot_bot_set_o ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X4 ) )
          = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3713_set__replicate__conv__if,axiom,
    ! [N: nat,X4: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X4 ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X4 ) )
          = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3714_set__replicate__conv__if,axiom,
    ! [N: nat,X4: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X4 ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X4 ) )
          = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3715_prod__decode__aux_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_3716_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3717_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3718_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3719_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3720_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3721_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3722_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8648204552663881920BT_int @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3723_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L234762979517870878al_nat @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3724_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3725_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_3726_the__elem__eq,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( the_elem_VEBT_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
      = X4 ) ).

% the_elem_eq
thf(fact_3727_the__elem__eq,axiom,
    ! [X4: real] :
      ( ( the_elem_real @ ( insert_real @ X4 @ bot_bot_set_real ) )
      = X4 ) ).

% the_elem_eq
thf(fact_3728_the__elem__eq,axiom,
    ! [X4: $o] :
      ( ( the_elem_o @ ( insert_o @ X4 @ bot_bot_set_o ) )
      = X4 ) ).

% the_elem_eq
thf(fact_3729_the__elem__eq,axiom,
    ! [X4: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
      = X4 ) ).

% the_elem_eq
thf(fact_3730_the__elem__eq,axiom,
    ! [X4: int] :
      ( ( the_elem_int @ ( insert_int @ X4 @ bot_bot_set_int ) )
      = X4 ) ).

% the_elem_eq
thf(fact_3731_sum__diff1_H__aux,axiom,
    ! [F5: set_VEBT_VEBT,I5: set_VEBT_VEBT,F: vEBT_VEBT > complex,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_complex ) ) )
          @ F5 )
       => ( ( ( member_VEBT_VEBT @ I @ I5 )
           => ( ( groups3521240112447731263omplex @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( minus_minus_complex @ ( groups3521240112447731263omplex @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_VEBT_VEBT @ I @ I5 )
           => ( ( groups3521240112447731263omplex @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( groups3521240112447731263omplex @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3732_sum__diff1_H__aux,axiom,
    ! [F5: set_complex,I5: set_complex,F: complex > complex,I: complex] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_complex ) ) )
          @ F5 )
       => ( ( ( member_complex @ I @ I5 )
           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_complex @ I @ I5 )
           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( groups808145749697022017omplex @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3733_sum__diff1_H__aux,axiom,
    ! [F5: set_Code_integer,I5: set_Code_integer,F: code_integer > complex,I: code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_complex ) ) )
          @ F5 )
       => ( ( ( member_Code_integer @ I @ I5 )
           => ( ( groups3262226078671967728omplex @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
              = ( minus_minus_complex @ ( groups3262226078671967728omplex @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_Code_integer @ I @ I5 )
           => ( ( groups3262226078671967728omplex @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
              = ( groups3262226078671967728omplex @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3734_sum__diff1_H__aux,axiom,
    ! [F5: set_real,I5: set_real,F: real > complex,I: real] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_complex ) ) )
          @ F5 )
       => ( ( ( member_real @ I @ I5 )
           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
              = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_real @ I @ I5 )
           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
              = ( groups5683813829254066239omplex @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3735_sum__diff1_H__aux,axiom,
    ! [F5: set_o,I5: set_o,F: $o > complex,I: $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_complex ) ) )
          @ F5 )
       => ( ( ( member_o @ I @ I5 )
           => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
              = ( minus_minus_complex @ ( groups3443914341975893411omplex @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_o @ I @ I5 )
           => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
              = ( groups3443914341975893411omplex @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3736_sum__diff1_H__aux,axiom,
    ! [F5: set_VEBT_VEBT,I5: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ F5 )
     => ( ( ord_le4337996190870823476T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_real ) ) )
          @ F5 )
       => ( ( ( member_VEBT_VEBT @ I @ I5 )
           => ( ( groups7811786883911161277T_real @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( minus_minus_real @ ( groups7811786883911161277T_real @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_VEBT_VEBT @ I @ I5 )
           => ( ( groups7811786883911161277T_real @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( groups7811786883911161277T_real @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3737_sum__diff1_H__aux,axiom,
    ! [F5: set_complex,I5: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_real ) ) )
          @ F5 )
       => ( ( ( member_complex @ I @ I5 )
           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_complex @ I @ I5 )
           => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( groups5737402329758386879x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3738_sum__diff1_H__aux,axiom,
    ! [F5: set_Code_integer,I5: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_real ) ) )
          @ F5 )
       => ( ( ( member_Code_integer @ I @ I5 )
           => ( ( groups9040905619451072238r_real @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
              = ( minus_minus_real @ ( groups9040905619451072238r_real @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_Code_integer @ I @ I5 )
           => ( ( groups9040905619451072238r_real @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
              = ( groups9040905619451072238r_real @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3739_sum__diff1_H__aux,axiom,
    ! [F5: set_real,I5: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ F5 )
     => ( ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_real ) ) )
          @ F5 )
       => ( ( ( member_real @ I @ I5 )
           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
              = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_real @ I @ I5 )
           => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
              = ( groups97945582718554045l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3740_sum__diff1_H__aux,axiom,
    ! [F5: set_o,I5: set_o,F: $o > real,I: $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I5 )
                & ( ( F @ I3 )
                 != zero_zero_real ) ) )
          @ F5 )
       => ( ( ( member_o @ I @ I5 )
           => ( ( groups627172608727702305o_real @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
              = ( minus_minus_real @ ( groups627172608727702305o_real @ F @ I5 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_o @ I @ I5 )
           => ( ( groups627172608727702305o_real @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
              = ( groups627172608727702305o_real @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_3741_is__singletonI,axiom,
    ! [X4: vEBT_VEBT] : ( is_sin24926331636114728T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ).

% is_singletonI
thf(fact_3742_is__singletonI,axiom,
    ! [X4: real] : ( is_singleton_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_3743_is__singletonI,axiom,
    ! [X4: $o] : ( is_singleton_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_3744_is__singletonI,axiom,
    ! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_3745_is__singletonI,axiom,
    ! [X4: int] : ( is_singleton_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_3746_sum__diff1_H,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > complex,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups3521240112447731263omplex @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_complex @ ( groups3521240112447731263omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups3521240112447731263omplex @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups3521240112447731263omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3747_sum__diff1_H,axiom,
    ! [I5: set_complex,F: complex > complex,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_complex @ I @ I5 )
         => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_complex @ I @ I5 )
         => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( groups808145749697022017omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3748_sum__diff1_H,axiom,
    ! [I5: set_Code_integer,F: code_integer > complex,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_Code_integer @ I @ I5 )
         => ( ( groups3262226078671967728omplex @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_complex @ ( groups3262226078671967728omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_Code_integer @ I @ I5 )
         => ( ( groups3262226078671967728omplex @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
            = ( groups3262226078671967728omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3749_sum__diff1_H,axiom,
    ! [I5: set_real,F: real > complex,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
            = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
            = ( groups5683813829254066239omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3750_sum__diff1_H,axiom,
    ! [I5: set_o,F: $o > complex,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_o @ I @ I5 )
         => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
            = ( minus_minus_complex @ ( groups3443914341975893411omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_o @ I @ I5 )
         => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I @ bot_bot_set_o ) ) )
            = ( groups3443914341975893411omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3751_sum__diff1_H,axiom,
    ! [I5: set_int,F: int > complex,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_int @ I @ I5 )
         => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I5 @ ( insert_int @ I @ bot_bot_set_int ) ) )
            = ( minus_minus_complex @ ( groups267424677133301183omplex @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_int @ I @ I5 )
         => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I5 @ ( insert_int @ I @ bot_bot_set_int ) ) )
            = ( groups267424677133301183omplex @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3752_sum__diff1_H,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7811786883911161277T_real @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_real @ ( groups7811786883911161277T_real @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7811786883911161277T_real @ F @ ( minus_5127226145743854075T_VEBT @ I5 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups7811786883911161277T_real @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3753_sum__diff1_H,axiom,
    ! [I5: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_complex @ I @ I5 )
         => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5737402329758386879x_real @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_complex @ I @ I5 )
         => ( ( groups5737402329758386879x_real @ F @ ( minus_811609699411566653omplex @ I5 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( groups5737402329758386879x_real @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3754_sum__diff1_H,axiom,
    ! [I5: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_Code_integer @ I @ I5 )
         => ( ( groups9040905619451072238r_real @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_real @ ( groups9040905619451072238r_real @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_Code_integer @ I @ I5 )
         => ( ( groups9040905619451072238r_real @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
            = ( groups9040905619451072238r_real @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3755_sum__diff1_H,axiom,
    ! [I5: set_real,F: real > real,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( F @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups97945582718554045l_real @ F @ I5 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups97945582718554045l_real @ F @ ( minus_minus_set_real @ I5 @ ( insert_real @ I @ bot_bot_set_real ) ) )
            = ( groups97945582718554045l_real @ F @ I5 ) ) ) ) ) ).

% sum_diff1'
thf(fact_3756_set__removeAll,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X4 @ Xs2 ) )
      = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_removeAll
thf(fact_3757_set__removeAll,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( set_real2 @ ( removeAll_real @ X4 @ Xs2 ) )
      = ( minus_minus_set_real @ ( set_real2 @ Xs2 ) @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ).

% set_removeAll
thf(fact_3758_set__removeAll,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( set_o2 @ ( removeAll_o @ X4 @ Xs2 ) )
      = ( minus_minus_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% set_removeAll
thf(fact_3759_set__removeAll,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( set_int2 @ ( removeAll_int @ X4 @ Xs2 ) )
      = ( minus_minus_set_int @ ( set_int2 @ Xs2 ) @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% set_removeAll
thf(fact_3760_set__removeAll,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( set_nat2 @ ( removeAll_nat @ X4 @ Xs2 ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% set_removeAll
thf(fact_3761_prod_Oinsert_H,axiom,
    ! [I5: set_o,P5: $o > real,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_o @ I @ I5 )
         => ( ( groups9056774472907672310o_real @ P5 @ ( insert_o @ I @ I5 ) )
            = ( groups9056774472907672310o_real @ P5 @ I5 ) ) )
        & ( ~ ( member_o @ I @ I5 )
         => ( ( groups9056774472907672310o_real @ P5 @ ( insert_o @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups9056774472907672310o_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3762_prod_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P5: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups5615669421813869842T_real @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( groups5615669421813869842T_real @ P5 @ I5 ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups5615669421813869842T_real @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups5615669421813869842T_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3763_prod_Oinsert_H,axiom,
    ! [I5: set_real,P5: real > real,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups1542684614274415016l_real @ P5 @ ( insert_real @ I @ I5 ) )
            = ( groups1542684614274415016l_real @ P5 @ I5 ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups1542684614274415016l_real @ P5 @ ( insert_real @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups1542684614274415016l_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3764_prod_Oinsert_H,axiom,
    ! [I5: set_nat,P5: nat > real,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_nat @ I @ I5 )
         => ( ( groups2144956460914285644t_real @ P5 @ ( insert_nat @ I @ I5 ) )
            = ( groups2144956460914285644t_real @ P5 @ I5 ) ) )
        & ( ~ ( member_nat @ I @ I5 )
         => ( ( groups2144956460914285644t_real @ P5 @ ( insert_nat @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups2144956460914285644t_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3765_prod_Oinsert_H,axiom,
    ! [I5: set_int,P5: int > real,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_int @ I @ I5 )
         => ( ( groups4331878035607307432t_real @ P5 @ ( insert_int @ I @ I5 ) )
            = ( groups4331878035607307432t_real @ P5 @ I5 ) ) )
        & ( ~ ( member_int @ I @ I5 )
         => ( ( groups4331878035607307432t_real @ P5 @ ( insert_int @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups4331878035607307432t_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3766_prod_Oinsert_H,axiom,
    ! [I5: set_complex,P5: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_complex @ I @ I5 )
         => ( ( groups5446087311781667882x_real @ P5 @ ( insert_complex @ I @ I5 ) )
            = ( groups5446087311781667882x_real @ P5 @ I5 ) ) )
        & ( ~ ( member_complex @ I @ I5 )
         => ( ( groups5446087311781667882x_real @ P5 @ ( insert_complex @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups5446087311781667882x_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3767_prod_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P5: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( member_Code_integer @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_real ) ) ) )
     => ( ( ( member_Code_integer @ I @ I5 )
         => ( ( groups7484263781351690841r_real @ P5 @ ( insert_Code_integer @ I @ I5 ) )
            = ( groups7484263781351690841r_real @ P5 @ I5 ) ) )
        & ( ~ ( member_Code_integer @ I @ I5 )
         => ( ( groups7484263781351690841r_real @ P5 @ ( insert_Code_integer @ I @ I5 ) )
            = ( times_times_real @ ( P5 @ I ) @ ( groups7484263781351690841r_real @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3768_prod_Oinsert_H,axiom,
    ! [I5: set_o,P5: $o > rat,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_rat ) ) ) )
     => ( ( ( member_o @ I @ I5 )
         => ( ( groups4688112016125141794_o_rat @ P5 @ ( insert_o @ I @ I5 ) )
            = ( groups4688112016125141794_o_rat @ P5 @ I5 ) ) )
        & ( ~ ( member_o @ I @ I5 )
         => ( ( groups4688112016125141794_o_rat @ P5 @ ( insert_o @ I @ I5 ) )
            = ( times_times_rat @ ( P5 @ I ) @ ( groups4688112016125141794_o_rat @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3769_prod_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P5: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_rat ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7596523497119011646BT_rat @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( groups7596523497119011646BT_rat @ P5 @ I5 ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7596523497119011646BT_rat @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( times_times_rat @ ( P5 @ I ) @ ( groups7596523497119011646BT_rat @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3770_prod_Oinsert_H,axiom,
    ! [I5: set_real,P5: real > rat,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( P5 @ X )
               != one_one_rat ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups6077134973956688596al_rat @ P5 @ ( insert_real @ I @ I5 ) )
            = ( groups6077134973956688596al_rat @ P5 @ I5 ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups6077134973956688596al_rat @ P5 @ ( insert_real @ I @ I5 ) )
            = ( times_times_rat @ ( P5 @ I ) @ ( groups6077134973956688596al_rat @ P5 @ I5 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_3771_sum_Oempty_H,axiom,
    ! [P5: real > complex] :
      ( ( groups5683813829254066239omplex @ P5 @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty'
thf(fact_3772_sum_Oempty_H,axiom,
    ! [P5: real > real] :
      ( ( groups97945582718554045l_real @ P5 @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty'
thf(fact_3773_sum_Oempty_H,axiom,
    ! [P5: real > rat] :
      ( ( groups3269169158384524137al_rat @ P5 @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty'
thf(fact_3774_sum_Oempty_H,axiom,
    ! [P5: real > nat] :
      ( ( groups3904299218471019873al_nat @ P5 @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty'
thf(fact_3775_sum_Oempty_H,axiom,
    ! [P5: real > int] :
      ( ( groups3901808747961969597al_int @ P5 @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty'
thf(fact_3776_sum_Oempty_H,axiom,
    ! [P5: $o > complex] :
      ( ( groups3443914341975893411omplex @ P5 @ bot_bot_set_o )
      = zero_zero_complex ) ).

% sum.empty'
thf(fact_3777_sum_Oempty_H,axiom,
    ! [P5: $o > real] :
      ( ( groups627172608727702305o_real @ P5 @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty'
thf(fact_3778_sum_Oempty_H,axiom,
    ! [P5: $o > rat] :
      ( ( groups3921277224699582669_o_rat @ P5 @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty'
thf(fact_3779_sum_Oempty_H,axiom,
    ! [P5: $o > nat] :
      ( ( groups4556407284786078405_o_nat @ P5 @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty'
thf(fact_3780_sum_Oempty_H,axiom,
    ! [P5: $o > int] :
      ( ( groups4553916814277028129_o_int @ P5 @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty'
thf(fact_3781_prod_Oempty_H,axiom,
    ! [P5: real > assn] :
      ( ( groups5834761644390034980l_assn @ P5 @ bot_bot_set_real )
      = one_one_assn ) ).

% prod.empty'
thf(fact_3782_prod_Oempty_H,axiom,
    ! [P5: real > real] :
      ( ( groups1542684614274415016l_real @ P5 @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty'
thf(fact_3783_prod_Oempty_H,axiom,
    ! [P5: real > rat] :
      ( ( groups6077134973956688596al_rat @ P5 @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty'
thf(fact_3784_prod_Oempty_H,axiom,
    ! [P5: real > nat] :
      ( ( groups6712265034043184332al_nat @ P5 @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty'
thf(fact_3785_prod_Oempty_H,axiom,
    ! [P5: real > int] :
      ( ( groups6709774563534134056al_int @ P5 @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty'
thf(fact_3786_prod_Oempty_H,axiom,
    ! [P5: $o > assn] :
      ( ( groups12625163056032370o_assn @ P5 @ bot_bot_set_o )
      = one_one_assn ) ).

% prod.empty'
thf(fact_3787_prod_Oempty_H,axiom,
    ! [P5: $o > real] :
      ( ( groups9056774472907672310o_real @ P5 @ bot_bot_set_o )
      = one_one_real ) ).

% prod.empty'
thf(fact_3788_prod_Oempty_H,axiom,
    ! [P5: $o > rat] :
      ( ( groups4688112016125141794_o_rat @ P5 @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty'
thf(fact_3789_prod_Oempty_H,axiom,
    ! [P5: $o > nat] :
      ( ( groups5323242076211637530_o_nat @ P5 @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty'
thf(fact_3790_prod_Oempty_H,axiom,
    ! [P5: $o > int] :
      ( ( groups5320751605702587254_o_int @ P5 @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty'
thf(fact_3791_listI__assn__finite,axiom,
    ! [I5: set_nat,A3: vEBT_VEBT > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ ( finite_finite_nat @ I5 )
     => ( ( vEBT_L1528199826722428489_VEBTi @ I5 @ A3 @ Xs2 @ Xsi )
        = bot_bot_assn ) ) ).

% listI_assn_finite
thf(fact_3792_sum_Oinsert_H,axiom,
    ! [I5: set_o,P5: $o > complex,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_o @ I @ I5 )
         => ( ( groups3443914341975893411omplex @ P5 @ ( insert_o @ I @ I5 ) )
            = ( groups3443914341975893411omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_o @ I @ I5 )
         => ( ( groups3443914341975893411omplex @ P5 @ ( insert_o @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups3443914341975893411omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3793_sum_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P5: vEBT_VEBT > complex,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups3521240112447731263omplex @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( groups3521240112447731263omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups3521240112447731263omplex @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups3521240112447731263omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3794_sum_Oinsert_H,axiom,
    ! [I5: set_real,P5: real > complex,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups5683813829254066239omplex @ P5 @ ( insert_real @ I @ I5 ) )
            = ( groups5683813829254066239omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups5683813829254066239omplex @ P5 @ ( insert_real @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups5683813829254066239omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3795_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P5: nat > complex,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_nat @ I @ I5 )
         => ( ( groups8515261248781899619omplex @ P5 @ ( insert_nat @ I @ I5 ) )
            = ( groups8515261248781899619omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_nat @ I @ I5 )
         => ( ( groups8515261248781899619omplex @ P5 @ ( insert_nat @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups8515261248781899619omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3796_sum_Oinsert_H,axiom,
    ! [I5: set_int,P5: int > complex,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_int @ I @ I5 )
         => ( ( groups267424677133301183omplex @ P5 @ ( insert_int @ I @ I5 ) )
            = ( groups267424677133301183omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_int @ I @ I5 )
         => ( ( groups267424677133301183omplex @ P5 @ ( insert_int @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups267424677133301183omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3797_sum_Oinsert_H,axiom,
    ! [I5: set_complex,P5: complex > complex,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_complex @ I @ I5 )
         => ( ( groups808145749697022017omplex @ P5 @ ( insert_complex @ I @ I5 ) )
            = ( groups808145749697022017omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_complex @ I @ I5 )
         => ( ( groups808145749697022017omplex @ P5 @ ( insert_complex @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups808145749697022017omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3798_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P5: code_integer > complex,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( member_Code_integer @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_Code_integer @ I @ I5 )
         => ( ( groups3262226078671967728omplex @ P5 @ ( insert_Code_integer @ I @ I5 ) )
            = ( groups3262226078671967728omplex @ P5 @ I5 ) ) )
        & ( ~ ( member_Code_integer @ I @ I5 )
         => ( ( groups3262226078671967728omplex @ P5 @ ( insert_Code_integer @ I @ I5 ) )
            = ( plus_plus_complex @ ( P5 @ I ) @ ( groups3262226078671967728omplex @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3799_sum_Oinsert_H,axiom,
    ! [I5: set_o,P5: $o > real,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_real ) ) ) )
     => ( ( ( member_o @ I @ I5 )
         => ( ( groups627172608727702305o_real @ P5 @ ( insert_o @ I @ I5 ) )
            = ( groups627172608727702305o_real @ P5 @ I5 ) ) )
        & ( ~ ( member_o @ I @ I5 )
         => ( ( groups627172608727702305o_real @ P5 @ ( insert_o @ I @ I5 ) )
            = ( plus_plus_real @ ( P5 @ I ) @ ( groups627172608727702305o_real @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3800_sum_Oinsert_H,axiom,
    ! [I5: set_VEBT_VEBT,P5: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_real ) ) ) )
     => ( ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7811786883911161277T_real @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( groups7811786883911161277T_real @ P5 @ I5 ) ) )
        & ( ~ ( member_VEBT_VEBT @ I @ I5 )
         => ( ( groups7811786883911161277T_real @ P5 @ ( insert_VEBT_VEBT @ I @ I5 ) )
            = ( plus_plus_real @ ( P5 @ I ) @ ( groups7811786883911161277T_real @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3801_sum_Oinsert_H,axiom,
    ! [I5: set_real,P5: real > real,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( P5 @ X )
               != zero_zero_real ) ) ) )
     => ( ( ( member_real @ I @ I5 )
         => ( ( groups97945582718554045l_real @ P5 @ ( insert_real @ I @ I5 ) )
            = ( groups97945582718554045l_real @ P5 @ I5 ) ) )
        & ( ~ ( member_real @ I @ I5 )
         => ( ( groups97945582718554045l_real @ P5 @ ( insert_real @ I @ I5 ) )
            = ( plus_plus_real @ ( P5 @ I ) @ ( groups97945582718554045l_real @ P5 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_3802_sum_Onon__neutral_H,axiom,
    ! [G: vEBT_VEBT > complex,I5: set_VEBT_VEBT] :
      ( ( groups3521240112447731263omplex @ G
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
      = ( groups3521240112447731263omplex @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3803_sum_Onon__neutral_H,axiom,
    ! [G: real > complex,I5: set_real] :
      ( ( groups5683813829254066239omplex @ G
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
      = ( groups5683813829254066239omplex @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3804_sum_Onon__neutral_H,axiom,
    ! [G: complex > complex,I5: set_complex] :
      ( ( groups808145749697022017omplex @ G
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
      = ( groups808145749697022017omplex @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3805_sum_Onon__neutral_H,axiom,
    ! [G: nat > complex,I5: set_nat] :
      ( ( groups8515261248781899619omplex @ G
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
      = ( groups8515261248781899619omplex @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3806_sum_Onon__neutral_H,axiom,
    ! [G: int > complex,I5: set_int] :
      ( ( groups267424677133301183omplex @ G
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
      = ( groups267424677133301183omplex @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3807_sum_Onon__neutral_H,axiom,
    ! [G: vEBT_VEBT > real,I5: set_VEBT_VEBT] :
      ( ( groups7811786883911161277T_real @ G
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
      = ( groups7811786883911161277T_real @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3808_sum_Onon__neutral_H,axiom,
    ! [G: real > real,I5: set_real] :
      ( ( groups97945582718554045l_real @ G
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
      = ( groups97945582718554045l_real @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3809_sum_Onon__neutral_H,axiom,
    ! [G: complex > real,I5: set_complex] :
      ( ( groups5737402329758386879x_real @ G
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
      = ( groups5737402329758386879x_real @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3810_sum_Onon__neutral_H,axiom,
    ! [G: nat > real,I5: set_nat] :
      ( ( groups8560362682196896993t_real @ G
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
      = ( groups8560362682196896993t_real @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3811_sum_Onon__neutral_H,axiom,
    ! [G: int > real,I5: set_int] :
      ( ( groups1523912220035142973t_real @ G
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
      = ( groups1523912220035142973t_real @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_3812_prod_Onon__neutral_H,axiom,
    ! [G: vEBT_VEBT > assn,I5: set_VEBT_VEBT] :
      ( ( groups8339234656476930702T_assn @ G
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != one_one_assn ) ) ) )
      = ( groups8339234656476930702T_assn @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3813_prod_Onon__neutral_H,axiom,
    ! [G: real > assn,I5: set_real] :
      ( ( groups5834761644390034980l_assn @ G
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != one_one_assn ) ) ) )
      = ( groups5834761644390034980l_assn @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3814_prod_Onon__neutral_H,axiom,
    ! [G: complex > assn,I5: set_complex] :
      ( ( groups8985371454280824230x_assn @ G
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != one_one_assn ) ) ) )
      = ( groups8985371454280824230x_assn @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3815_prod_Onon__neutral_H,axiom,
    ! [G: nat > assn,I5: set_nat] :
      ( ( groups3916077543535352520t_assn @ G
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != one_one_assn ) ) ) )
      = ( groups3916077543535352520t_assn @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3816_prod_Onon__neutral_H,axiom,
    ! [G: int > assn,I5: set_int] :
      ( ( groups4891613008741529892t_assn @ G
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != one_one_assn ) ) ) )
      = ( groups4891613008741529892t_assn @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3817_prod_Onon__neutral_H,axiom,
    ! [G: vEBT_VEBT > real,I5: set_VEBT_VEBT] :
      ( ( groups5615669421813869842T_real @ G
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
      = ( groups5615669421813869842T_real @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3818_prod_Onon__neutral_H,axiom,
    ! [G: real > real,I5: set_real] :
      ( ( groups1542684614274415016l_real @ G
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
      = ( groups1542684614274415016l_real @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3819_prod_Onon__neutral_H,axiom,
    ! [G: complex > real,I5: set_complex] :
      ( ( groups5446087311781667882x_real @ G
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
      = ( groups5446087311781667882x_real @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3820_prod_Onon__neutral_H,axiom,
    ! [G: nat > real,I5: set_nat] :
      ( ( groups2144956460914285644t_real @ G
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
      = ( groups2144956460914285644t_real @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3821_prod_Onon__neutral_H,axiom,
    ! [G: int > real,I5: set_int] :
      ( ( groups4331878035607307432t_real @ G
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
      = ( groups4331878035607307432t_real @ G @ I5 ) ) ).

% prod.non_neutral'
thf(fact_3822_is__singleton__the__elem,axiom,
    ( is_sin24926331636114728T_VEBT
    = ( ^ [A5: set_VEBT_VEBT] :
          ( A5
          = ( insert_VEBT_VEBT @ ( the_elem_VEBT_VEBT @ A5 ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% is_singleton_the_elem
thf(fact_3823_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
          ( A5
          = ( insert_real @ ( the_elem_real @ A5 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_3824_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_3825_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_3826_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
          ( A5
          = ( insert_int @ ( the_elem_int @ A5 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_3827_is__singletonI_H,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( A3 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ( member_VEBT_VEBT @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_sin24926331636114728T_VEBT @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3828_is__singletonI_H,axiom,
    ! [A3: set_set_nat] :
      ( ( A3 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ( member_set_nat @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_set_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3829_is__singletonI_H,axiom,
    ! [A3: set_real] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X3: real,Y3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ( member_real @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_real @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3830_is__singletonI_H,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ( member_o @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3831_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ( member_nat @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3832_is__singletonI_H,axiom,
    ! [A3: set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [X3: int,Y3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ( member_int @ Y3 @ A3 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_int @ A3 ) ) ) ).

% is_singletonI'
thf(fact_3833_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups8560362682196896993t_real
          @ ^ [I3: nat] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I5 ) @ ( groups8560362682196896993t_real @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3834_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups1523912220035142973t_real
          @ ^ [I3: int] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I5 ) @ ( groups1523912220035142973t_real @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3835_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( groups5737402329758386879x_real
          @ ^ [I3: complex] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I5 ) @ ( groups5737402329758386879x_real @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3836_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups9040905619451072238r_real
          @ ^ [I3: code_integer] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_real @ ( groups9040905619451072238r_real @ G @ I5 ) @ ( groups9040905619451072238r_real @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3837_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups1351286907653491341at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I5 ) @ ( groups1351286907653491341at_rat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3838_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups2350640619554545897nt_rat
          @ ^ [I3: int] : ( plus_plus_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I5 ) @ ( groups2350640619554545897nt_rat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3839_sum_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( groups2276542476275365739ex_rat
          @ ^ [I3: complex] : ( plus_plus_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups2276542476275365739ex_rat @ G @ I5 ) @ ( groups2276542476275365739ex_rat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3840_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > rat,H2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups8878813951405302554er_rat
          @ ^ [I3: code_integer] : ( plus_plus_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups8878813951405302554er_rat @ G @ I5 ) @ ( groups8878813951405302554er_rat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3841_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > nat,H2: nat > nat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups1986416967739987077at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I5 ) @ ( groups1986416967739987077at_nat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3842_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups2985770679641041633nt_nat
          @ ^ [I3: int] : ( plus_plus_nat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I5 ) @ ( groups2985770679641041633nt_nat @ H2 @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_3843_length__removeAll__less__eq,axiom,
    ! [X4: real,Xs2: list_real] : ( ord_less_eq_nat @ ( size_size_list_real @ ( removeAll_real @ X4 @ Xs2 ) ) @ ( size_size_list_real @ Xs2 ) ) ).

% length_removeAll_less_eq
thf(fact_3844_length__removeAll__less__eq,axiom,
    ! [X4: $o,Xs2: list_o] : ( ord_less_eq_nat @ ( size_size_list_o @ ( removeAll_o @ X4 @ Xs2 ) ) @ ( size_size_list_o @ Xs2 ) ) ).

% length_removeAll_less_eq
thf(fact_3845_length__removeAll__less__eq,axiom,
    ! [X4: nat,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X4 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).

% length_removeAll_less_eq
thf(fact_3846_length__removeAll__less__eq,axiom,
    ! [X4: int,Xs2: list_int] : ( ord_less_eq_nat @ ( size_size_list_int @ ( removeAll_int @ X4 @ Xs2 ) ) @ ( size_size_list_int @ Xs2 ) ) ).

% length_removeAll_less_eq
thf(fact_3847_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups2144956460914285644t_real
          @ ^ [I3: nat] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_real @ ( groups2144956460914285644t_real @ G @ I5 ) @ ( groups2144956460914285644t_real @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3848_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups4331878035607307432t_real
          @ ^ [I3: int] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_real @ ( groups4331878035607307432t_real @ G @ I5 ) @ ( groups4331878035607307432t_real @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3849_prod_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( groups5446087311781667882x_real
          @ ^ [I3: complex] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_real @ ( groups5446087311781667882x_real @ G @ I5 ) @ ( groups5446087311781667882x_real @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3850_prod_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups7484263781351690841r_real
          @ ^ [I3: code_integer] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_real @ ( groups7484263781351690841r_real @ G @ I5 ) @ ( groups7484263781351690841r_real @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3851_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups4112464933559648120at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_rat @ ( groups4112464933559648120at_rat @ G @ I5 ) @ ( groups4112464933559648120at_rat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3852_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups5111818645460702676nt_rat
          @ ^ [I3: int] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_rat @ ( groups5111818645460702676nt_rat @ G @ I5 ) @ ( groups5111818645460702676nt_rat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3853_prod_Odistrib__triv_H,axiom,
    ! [I5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( groups6458467974770906710ex_rat
          @ ^ [I3: complex] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_rat @ ( groups6458467974770906710ex_rat @ G @ I5 ) @ ( groups6458467974770906710ex_rat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3854_prod_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > rat,H2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups7390404786021641221er_rat
          @ ^ [I3: code_integer] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_rat @ ( groups7390404786021641221er_rat @ G @ I5 ) @ ( groups7390404786021641221er_rat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3855_prod_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > nat,H2: nat > nat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups4747594993646143856at_nat
          @ ^ [I3: nat] : ( times_times_nat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_nat @ ( groups4747594993646143856at_nat @ G @ I5 ) @ ( groups4747594993646143856at_nat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3856_prod_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups5746948705547198412nt_nat
          @ ^ [I3: int] : ( times_times_nat @ ( G @ I3 ) @ ( H2 @ I3 ) )
          @ I5 )
        = ( times_times_nat @ ( groups5746948705547198412nt_nat @ G @ I5 ) @ ( groups5746948705547198412nt_nat @ H2 @ I5 ) ) ) ) ).

% prod.distrib_triv'
thf(fact_3857_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3521240112447731263omplex @ G @ T3 )
            = ( groups3521240112447731263omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3858_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5683813829254066239omplex @ G @ T3 )
            = ( groups5683813829254066239omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3859_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( 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 ) ) )
         => ( ( groups7811786883911161277T_real @ G @ T3 )
            = ( groups7811786883911161277T_real @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3860_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > real,H2: real > real] :
      ( ( 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 ) ) )
         => ( ( groups97945582718554045l_real @ G @ T3 )
            = ( groups97945582718554045l_real @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3861_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( 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 ) ) )
         => ( ( groups7132981355262091625BT_rat @ G @ T3 )
            = ( groups7132981355262091625BT_rat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3862_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > rat,H2: real > rat] :
      ( ( 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 ) ) )
         => ( ( groups3269169158384524137al_rat @ G @ T3 )
            = ( groups3269169158384524137al_rat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3863_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > nat,H2: vEBT_VEBT > nat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7768111415348587361BT_nat @ G @ T3 )
            = ( groups7768111415348587361BT_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3864_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3904299218471019873al_nat @ G @ T3 )
            = ( groups3904299218471019873al_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3865_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > int,H2: vEBT_VEBT > int] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7765620944839537085BT_int @ G @ T3 )
            = ( groups7765620944839537085BT_int @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3866_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > int,H2: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3901808747961969597al_int @ G @ T3 )
            = ( groups3901808747961969597al_int @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_3867_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > complex,G: vEBT_VEBT > complex] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_complex ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3521240112447731263omplex @ G @ S3 )
            = ( groups3521240112447731263omplex @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3868_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > complex,G: real > complex] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_complex ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5683813829254066239omplex @ G @ S3 )
            = ( groups5683813829254066239omplex @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3869_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_real ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7811786883911161277T_real @ G @ S3 )
            = ( groups7811786883911161277T_real @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3870_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > real,G: real > real] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_real ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups97945582718554045l_real @ G @ S3 )
            = ( groups97945582718554045l_real @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3871_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_rat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7132981355262091625BT_rat @ G @ S3 )
            = ( groups7132981355262091625BT_rat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3872_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > rat,G: real > rat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_rat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3269169158384524137al_rat @ G @ S3 )
            = ( groups3269169158384524137al_rat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3873_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_nat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7768111415348587361BT_nat @ G @ S3 )
            = ( groups7768111415348587361BT_nat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3874_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > nat,G: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_nat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3904299218471019873al_nat @ G @ S3 )
            = ( groups3904299218471019873al_nat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3875_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_int ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7765620944839537085BT_int @ G @ S3 )
            = ( groups7765620944839537085BT_int @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3876_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > int,G: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = zero_zero_int ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups3901808747961969597al_int @ G @ S3 )
            = ( groups3901808747961969597al_int @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_3877_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > complex] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ( groups8515261248781899619omplex @ G @ T3 )
          = ( groups8515261248781899619omplex @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3878_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_real ) )
       => ( ( groups8560362682196896993t_real @ G @ T3 )
          = ( groups8560362682196896993t_real @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3879_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups1351286907653491341at_rat @ G @ T3 )
          = ( groups1351286907653491341at_rat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3880_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups1986416967739987077at_nat @ G @ T3 )
          = ( groups1986416967739987077at_nat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3881_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups1983926497230936801at_int @ G @ T3 )
          = ( groups1983926497230936801at_int @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3882_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > complex] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ( groups267424677133301183omplex @ G @ T3 )
          = ( groups267424677133301183omplex @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3883_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_real ) )
       => ( ( groups1523912220035142973t_real @ G @ T3 )
          = ( groups1523912220035142973t_real @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3884_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups2350640619554545897nt_rat @ G @ T3 )
          = ( groups2350640619554545897nt_rat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3885_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups2985770679641041633nt_nat @ G @ T3 )
          = ( groups2985770679641041633nt_nat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3886_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups2983280209131991357nt_int @ G @ T3 )
          = ( groups2983280209131991357nt_int @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_3887_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > complex] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ( groups8515261248781899619omplex @ G @ S3 )
          = ( groups8515261248781899619omplex @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3888_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_real ) )
       => ( ( groups8560362682196896993t_real @ G @ S3 )
          = ( groups8560362682196896993t_real @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3889_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups1351286907653491341at_rat @ G @ S3 )
          = ( groups1351286907653491341at_rat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3890_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups1986416967739987077at_nat @ G @ S3 )
          = ( groups1986416967739987077at_nat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3891_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups1983926497230936801at_int @ G @ S3 )
          = ( groups1983926497230936801at_int @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3892_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > complex] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_complex ) )
       => ( ( groups267424677133301183omplex @ G @ S3 )
          = ( groups267424677133301183omplex @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3893_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_real ) )
       => ( ( groups1523912220035142973t_real @ G @ S3 )
          = ( groups1523912220035142973t_real @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3894_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups2350640619554545897nt_rat @ G @ S3 )
          = ( groups2350640619554545897nt_rat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3895_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups2985770679641041633nt_nat @ G @ S3 )
          = ( groups2985770679641041633nt_nat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3896_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups2983280209131991357nt_int @ G @ S3 )
          = ( groups2983280209131991357nt_int @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_3897_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > vEBT_VEBTi > assn,A8: vEBT_VEBTi > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( 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 @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                          = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L886525131989349516_VEBTi @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3898_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > vEBT_VEBT > assn,A8: vEBT_VEBTi > vEBT_VEBT > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( 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 @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3899_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBT > vEBT_VEBT > assn,A8: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L3204528365124325536T_VEBT @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3900_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > real > assn,A8: vEBT_VEBTi > real > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( 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 @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_real @ Xsi @ I2 )
                          = ( nth_real @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L7728200936804140803i_real @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L7728200936804140803i_real @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3901_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBT > real > assn,A8: vEBT_VEBT > real > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                        & ( ( nth_real @ Xsi @ I2 )
                          = ( nth_real @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L4281036506115550016T_real @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L4281036506115550016T_real @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3902_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > $o > assn,A8: vEBT_VEBTi > $o > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( 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 @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_o @ Xsi @ I2 )
                          = ( nth_o @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3903_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBT > $o > assn,A8: vEBT_VEBT > $o > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                        & ( ( nth_o @ Xsi @ I2 )
                          = ( nth_o @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L7058566406413635588VEBT_o @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L7058566406413635588VEBT_o @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3904_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > nat > assn,A8: vEBT_VEBTi > nat > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_nat @ Xsi @ I2 )
                          = ( nth_nat @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L2809031099982602151Ti_nat @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3905_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBT > nat > assn,A8: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I2: nat] :
                  ( ( member_nat @ I2 @ I5 )
                 => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                        & ( ( nth_nat @ Xsi @ I2 )
                          = ( nth_nat @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L8650695023172932196BT_nat @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3906_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A3: vEBT_VEBTi > int > assn,A8: vEBT_VEBTi > int > assn,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A3 = A8 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( 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 @ Xs2 ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                        & ( ( nth_int @ Xsi @ I2 )
                          = ( nth_int @ Xsi2 @ I2 ) ) ) ) ) )
             => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_L2806540629473551875Ti_int @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_3907_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A3: vEBT_VEBTi > vEBT_VEBTi > assn,A8: vEBT_VEBTi > vEBT_VEBTi > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( 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 @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBTi @ Xsi @ I2 )
                        = ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_VEBT_VEBTi @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L886525131989349516_VEBTi @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3908_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A3: vEBT_VEBTi > vEBT_VEBT > assn,A8: vEBT_VEBTi > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( 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 @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3909_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A3: vEBT_VEBT > vEBT_VEBT > assn,A8: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I2 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBT @ Xs4 @ I2 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L3204528365124325536T_VEBT @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3910_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A3: vEBT_VEBTi > real > assn,A8: vEBT_VEBTi > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( 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 @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_real @ Xsi @ I2 )
                        = ( nth_real @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_real @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_real @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L7728200936804140803i_real @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L7728200936804140803i_real @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3911_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A3: vEBT_VEBT > real > assn,A8: vEBT_VEBT > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                      & ( ( nth_real @ Xsi @ I2 )
                        = ( nth_real @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_real @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBT @ Xs4 @ I2 ) @ ( nth_real @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L4281036506115550016T_real @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L4281036506115550016T_real @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3912_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o,A3: vEBT_VEBTi > $o > assn,A8: vEBT_VEBTi > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( 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 @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_o @ Xsi @ I2 )
                        = ( nth_o @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_o @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_o @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3913_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o,A3: vEBT_VEBT > $o > assn,A8: vEBT_VEBT > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                      & ( ( nth_o @ Xsi @ I2 )
                        = ( nth_o @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_o @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBT @ Xs4 @ I2 ) @ ( nth_o @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L7058566406413635588VEBT_o @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L7058566406413635588VEBT_o @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3914_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat,A3: vEBT_VEBTi > nat > assn,A8: vEBT_VEBTi > nat > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_nat @ Xsi @ I2 )
                        = ( nth_nat @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_nat @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_nat @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L2809031099982602151Ti_nat @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3915_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A3: vEBT_VEBT > nat > assn,A8: vEBT_VEBT > nat > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I2 ) )
                      & ( ( nth_nat @ Xsi @ I2 )
                        = ( nth_nat @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_nat @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBT @ Xs4 @ I2 ) @ ( nth_nat @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L8650695023172932196BT_nat @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3916_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs2: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int,A3: vEBT_VEBTi > int > assn,A8: vEBT_VEBTi > int > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
          = ( 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 @ Xs2 ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I2 ) )
                      & ( ( nth_int @ Xsi @ I2 )
                        = ( nth_int @ Xsi2 @ I2 ) )
                      & ( ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) @ ( nth_int @ Xsi @ I2 ) )
                        = ( A8 @ ( nth_VEBT_VEBTi @ Xs4 @ I2 ) @ ( nth_int @ Xsi2 @ I2 ) ) ) ) ) ) )
           => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_L2806540629473551875Ti_int @ I6 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_3917_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ( groups3916077543535352520t_assn @ G @ S3 )
          = ( groups3916077543535352520t_assn @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3918_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ( groups2144956460914285644t_real @ G @ S3 )
          = ( groups2144956460914285644t_real @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3919_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ( groups4112464933559648120at_rat @ G @ S3 )
          = ( groups4112464933559648120at_rat @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3920_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ( groups4747594993646143856at_nat @ G @ S3 )
          = ( groups4747594993646143856at_nat @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3921_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ( groups4745104523137093580at_int @ G @ S3 )
          = ( groups4745104523137093580at_int @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3922_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > assn] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ( groups4891613008741529892t_assn @ G @ S3 )
          = ( groups4891613008741529892t_assn @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3923_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ( groups4331878035607307432t_real @ G @ S3 )
          = ( groups4331878035607307432t_real @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3924_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ( groups5111818645460702676nt_rat @ G @ S3 )
          = ( groups5111818645460702676nt_rat @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3925_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ( groups5746948705547198412nt_nat @ G @ S3 )
          = ( groups5746948705547198412nt_nat @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3926_prod_Omono__neutral__left_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ( groups5744458235038148136nt_int @ G @ S3 )
          = ( groups5744458235038148136nt_int @ G @ T3 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_3927_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ( groups3916077543535352520t_assn @ G @ T3 )
          = ( groups3916077543535352520t_assn @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3928_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ( groups2144956460914285644t_real @ G @ T3 )
          = ( groups2144956460914285644t_real @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3929_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ( groups4112464933559648120at_rat @ G @ T3 )
          = ( groups4112464933559648120at_rat @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3930_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ( groups4747594993646143856at_nat @ G @ T3 )
          = ( groups4747594993646143856at_nat @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3931_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ( groups4745104523137093580at_int @ G @ T3 )
          = ( groups4745104523137093580at_int @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3932_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > assn] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ( groups4891613008741529892t_assn @ G @ T3 )
          = ( groups4891613008741529892t_assn @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3933_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ( groups4331878035607307432t_real @ G @ T3 )
          = ( groups4331878035607307432t_real @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3934_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ( groups5111818645460702676nt_rat @ G @ T3 )
          = ( groups5111818645460702676nt_rat @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3935_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ( groups5746948705547198412nt_nat @ G @ T3 )
          = ( groups5746948705547198412nt_nat @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3936_prod_Omono__neutral__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ( groups5744458235038148136nt_int @ G @ T3 )
          = ( groups5744458235038148136nt_int @ G @ S3 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_3937_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > assn,G: vEBT_VEBT > assn] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_assn ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8339234656476930702T_assn @ G @ S3 )
            = ( groups8339234656476930702T_assn @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3938_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > assn,G: real > assn] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_assn ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5834761644390034980l_assn @ G @ S3 )
            = ( groups5834761644390034980l_assn @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3939_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_real ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5615669421813869842T_real @ G @ S3 )
            = ( groups5615669421813869842T_real @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3940_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > real,G: real > real] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_real ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups1542684614274415016l_real @ G @ S3 )
            = ( groups1542684614274415016l_real @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3941_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_rat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7596523497119011646BT_rat @ G @ S3 )
            = ( groups7596523497119011646BT_rat @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3942_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > rat,G: real > rat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_rat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6077134973956688596al_rat @ G @ S3 )
            = ( groups6077134973956688596al_rat @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3943_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_nat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8231653557205507382BT_nat @ G @ S3 )
            = ( groups8231653557205507382BT_nat @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3944_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > nat,G: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_nat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6712265034043184332al_nat @ G @ S3 )
            = ( groups6712265034043184332al_nat @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3945_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,H2: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_int ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8229163086696457106BT_int @ G @ S3 )
            = ( groups8229163086696457106BT_int @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3946_prod_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > int,G: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I2 )
              = one_one_int ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6709774563534134056al_int @ G @ S3 )
            = ( groups6709774563534134056al_int @ H2 @ T3 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_3947_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > assn,H2: vEBT_VEBT > assn] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8339234656476930702T_assn @ G @ T3 )
            = ( groups8339234656476930702T_assn @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3948_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > assn,H2: real > assn] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_assn ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5834761644390034980l_assn @ G @ T3 )
            = ( groups5834761644390034980l_assn @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3949_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups5615669421813869842T_real @ G @ T3 )
            = ( groups5615669421813869842T_real @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3950_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > real,H2: real > real] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_real ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups1542684614274415016l_real @ G @ T3 )
            = ( groups1542684614274415016l_real @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3951_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups7596523497119011646BT_rat @ G @ T3 )
            = ( groups7596523497119011646BT_rat @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3952_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > rat,H2: real > rat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_rat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6077134973956688596al_rat @ G @ T3 )
            = ( groups6077134973956688596al_rat @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3953_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > nat,H2: vEBT_VEBT > nat] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8231653557205507382BT_nat @ G @ T3 )
            = ( groups8231653557205507382BT_nat @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3954_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_nat ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6712265034043184332al_nat @ G @ T3 )
            = ( groups6712265034043184332al_nat @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3955_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_VEBT_VEBT,T3: set_VEBT_VEBT,G: vEBT_VEBT > int,H2: vEBT_VEBT > int] :
      ( ( ord_le4337996190870823476T_VEBT @ S3 @ T3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups8229163086696457106BT_int @ G @ T3 )
            = ( groups8229163086696457106BT_int @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3956_prod_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > int,H2: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X3 )
              = one_one_int ) )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ S3 )
             => ( ( G @ X3 )
                = ( H2 @ X3 ) ) )
         => ( ( groups6709774563534134056al_int @ G @ T3 )
            = ( groups6709774563534134056al_int @ H2 @ S3 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_3957_sum_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups3521240112447731263omplex
            @ ^ [I3: vEBT_VEBT] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups3521240112447731263omplex @ G @ I5 ) @ ( groups3521240112447731263omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3958_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups5683813829254066239omplex
            @ ^ [I3: real] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups5683813829254066239omplex @ G @ I5 ) @ ( groups5683813829254066239omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3959_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups8515261248781899619omplex
            @ ^ [I3: nat] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups8515261248781899619omplex @ G @ I5 ) @ ( groups8515261248781899619omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3960_sum_Odistrib_H,axiom,
    ! [I5: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups267424677133301183omplex
            @ ^ [I3: int] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups267424677133301183omplex @ G @ I5 ) @ ( groups267424677133301183omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3961_sum_Odistrib_H,axiom,
    ! [I5: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups808145749697022017omplex
            @ ^ [I3: complex] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups808145749697022017omplex @ G @ I5 ) @ ( groups808145749697022017omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3962_sum_Odistrib_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > complex,H2: code_integer > complex] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( member_Code_integer @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_complex ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_complex ) ) ) )
       => ( ( groups3262226078671967728omplex
            @ ^ [I3: code_integer] : ( plus_plus_complex @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_complex @ ( groups3262226078671967728omplex @ G @ I5 ) @ ( groups3262226078671967728omplex @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3963_sum_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_real ) ) ) )
       => ( ( groups7811786883911161277T_real
            @ ^ [I3: vEBT_VEBT] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_real @ ( groups7811786883911161277T_real @ G @ I5 ) @ ( groups7811786883911161277T_real @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3964_sum_Odistrib_H,axiom,
    ! [I5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_real ) ) ) )
       => ( ( groups97945582718554045l_real
            @ ^ [I3: real] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_real @ ( groups97945582718554045l_real @ G @ I5 ) @ ( groups97945582718554045l_real @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3965_sum_Odistrib_H,axiom,
    ! [I5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_real ) ) ) )
       => ( ( groups8560362682196896993t_real
            @ ^ [I3: nat] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I5 ) @ ( groups8560362682196896993t_real @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3966_sum_Odistrib_H,axiom,
    ! [I5: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ I5 )
                & ( ( H2 @ X )
                 != zero_zero_real ) ) ) )
       => ( ( groups1523912220035142973t_real
            @ ^ [I3: int] : ( plus_plus_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( plus_plus_real @ ( groups1523912220035142973t_real @ G @ I5 ) @ ( groups1523912220035142973t_real @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib'
thf(fact_3967_length__removeAll__less,axiom,
    ! [X4: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X4 @ Xs2 ) ) @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3968_length__removeAll__less,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X4 @ Xs2 ) ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3969_length__removeAll__less,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X4 @ Xs2 ) ) @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3970_length__removeAll__less,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X4 @ Xs2 ) ) @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3971_length__removeAll__less,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X4 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3972_length__removeAll__less,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X4 @ Xs2 ) ) @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_3973_prod_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups5615669421813869842T_real
            @ ^ [I3: vEBT_VEBT] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups5615669421813869842T_real @ G @ I5 ) @ ( groups5615669421813869842T_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3974_prod_Odistrib_H,axiom,
    ! [I5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups1542684614274415016l_real
            @ ^ [I3: real] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups1542684614274415016l_real @ G @ I5 ) @ ( groups1542684614274415016l_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3975_prod_Odistrib_H,axiom,
    ! [I5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups2144956460914285644t_real
            @ ^ [I3: nat] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups2144956460914285644t_real @ G @ I5 ) @ ( groups2144956460914285644t_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3976_prod_Odistrib_H,axiom,
    ! [I5: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups4331878035607307432t_real
            @ ^ [I3: int] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups4331878035607307432t_real @ G @ I5 ) @ ( groups4331878035607307432t_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3977_prod_Odistrib_H,axiom,
    ! [I5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups5446087311781667882x_real
            @ ^ [I3: complex] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups5446087311781667882x_real @ G @ I5 ) @ ( groups5446087311781667882x_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3978_prod_Odistrib_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( member_Code_integer @ X @ I5 )
              & ( ( G @ X )
               != one_one_real ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_real ) ) ) )
       => ( ( groups7484263781351690841r_real
            @ ^ [I3: code_integer] : ( times_times_real @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_real @ ( groups7484263781351690841r_real @ G @ I5 ) @ ( groups7484263781351690841r_real @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3979_prod_Odistrib_H,axiom,
    ! [I5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ I5 )
              & ( ( G @ X )
               != one_one_rat ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_rat ) ) ) )
       => ( ( groups7596523497119011646BT_rat
            @ ^ [I3: vEBT_VEBT] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_rat @ ( groups7596523497119011646BT_rat @ G @ I5 ) @ ( groups7596523497119011646BT_rat @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3980_prod_Odistrib_H,axiom,
    ! [I5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ I5 )
              & ( ( G @ X )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_rat ) ) ) )
       => ( ( groups6077134973956688596al_rat
            @ ^ [I3: real] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_rat @ ( groups6077134973956688596al_rat @ G @ I5 ) @ ( groups6077134973956688596al_rat @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3981_prod_Odistrib_H,axiom,
    ! [I5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I5 )
              & ( ( G @ X )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_rat ) ) ) )
       => ( ( groups4112464933559648120at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_rat @ ( groups4112464933559648120at_rat @ G @ I5 ) @ ( groups4112464933559648120at_rat @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3982_prod_Odistrib_H,axiom,
    ! [I5: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ I5 )
              & ( ( G @ X )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ I5 )
                & ( ( H2 @ X )
                 != one_one_rat ) ) ) )
       => ( ( groups5111818645460702676nt_rat
            @ ^ [I3: int] : ( times_times_rat @ ( G @ I3 ) @ ( H2 @ I3 ) )
            @ I5 )
          = ( times_times_rat @ ( groups5111818645460702676nt_rat @ G @ I5 ) @ ( groups5111818645460702676nt_rat @ H2 @ I5 ) ) ) ) ) ).

% prod.distrib'
thf(fact_3983_is__singleton__def,axiom,
    ( is_sin24926331636114728T_VEBT
    = ( ^ [A5: set_VEBT_VEBT] :
        ? [X: vEBT_VEBT] :
          ( A5
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% is_singleton_def
thf(fact_3984_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
        ? [X: real] :
          ( A5
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_3985_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X: $o] :
          ( A5
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_3986_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X: nat] :
          ( A5
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_3987_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
        ? [X: int] :
          ( A5
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_3988_is__singletonE,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( is_sin24926331636114728T_VEBT @ A3 )
     => ~ ! [X3: vEBT_VEBT] :
            ( A3
           != ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% is_singletonE
thf(fact_3989_is__singletonE,axiom,
    ! [A3: set_real] :
      ( ( is_singleton_real @ A3 )
     => ~ ! [X3: real] :
            ( A3
           != ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_3990_is__singletonE,axiom,
    ! [A3: set_o] :
      ( ( is_singleton_o @ A3 )
     => ~ ! [X3: $o] :
            ( A3
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_3991_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( is_singleton_nat @ A3 )
     => ~ ! [X3: nat] :
            ( A3
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_3992_is__singletonE,axiom,
    ! [A3: set_int] :
      ( ( is_singleton_int @ A3 )
     => ~ ! [X3: int] :
            ( A3
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_3993_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3994_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3995_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3996_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3997_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3998_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_3999_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8648204552663881920BT_int @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_4000_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L234762979517870878al_nat @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_4001_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_4002_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_4003_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > nat > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4004_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4005_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBT > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4006_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > int > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4007_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4008_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4009_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: vEBT_VEBT > int > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8648204552663881920BT_int @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4010_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: real > nat > assn,Xs2: list_real,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4011_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: real > vEBT_VEBTi > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4012_listI__assn__reinsert_H,axiom,
    ! [P: assn,A3: real > vEBT_VEBT > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_4013_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBTi > nat > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4014_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4015_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBT > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4016_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBTi > int > assn,Xs2: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4017_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4018_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4019_listI__assn__reinsert,axiom,
    ! [P: assn,A3: vEBT_VEBT > int > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8648204552663881920BT_int @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4020_listI__assn__reinsert,axiom,
    ! [P: assn,A3: real > nat > assn,Xs2: list_real,I: nat,Xsi: list_nat,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4021_listI__assn__reinsert,axiom,
    ! [P: assn,A3: real > vEBT_VEBTi > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4022_listI__assn__reinsert,axiom,
    ! [P: assn,A3: real > vEBT_VEBT > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_4023_in__set__product__lists__length,axiom,
    ! [Xs2: list_real,Xss: list_list_real] :
      ( ( member_list_real @ Xs2 @ ( set_list_real2 @ ( product_lists_real @ Xss ) ) )
     => ( ( size_size_list_real @ Xs2 )
        = ( size_s6660260683639930848t_real @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_4024_in__set__product__lists__length,axiom,
    ! [Xs2: list_o,Xss: list_list_o] :
      ( ( member_list_o @ Xs2 @ ( set_list_o2 @ ( product_lists_o @ Xss ) ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_s2710708370519433104list_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_4025_in__set__product__lists__length,axiom,
    ! [Xs2: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_4026_in__set__product__lists__length,axiom,
    ! [Xs2: list_int,Xss: list_list_int] :
      ( ( member_list_int @ Xs2 @ ( set_list_int2 @ ( product_lists_int @ Xss ) ) )
     => ( ( size_size_list_int @ Xs2 )
        = ( size_s533118279054570080st_int @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_4027_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_4028_Id__on__set,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( id_on_VEBT_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( set_Pr9182192707038809660T_VEBT
        @ ( map_VE1720758354293053111T_VEBT
          @ ^ [X: vEBT_VEBT] : ( produc537772716801021591T_VEBT @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4029_Id__on__set,axiom,
    ! [Xs2: list_real] :
      ( ( id_on_real @ ( set_real2 @ Xs2 ) )
      = ( set_Pr5999470521830281550l_real
        @ ( map_re3533924840734919879l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4030_Id__on__set,axiom,
    ! [Xs2: list_o] :
      ( ( id_on_o @ ( set_o2 @ Xs2 ) )
      = ( set_Product_prod_o_o2
        @ ( map_o_3702434973371374163od_o_o
          @ ^ [X: $o] : ( product_Pair_o_o @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4031_Id__on__set,axiom,
    ! [Xs2: list_nat] :
      ( ( id_on_nat @ ( set_nat2 @ Xs2 ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4032_Id__on__set,axiom,
    ! [Xs2: list_int] :
      ( ( id_on_int @ ( set_int2 @ Xs2 ) )
      = ( set_Pr2470121279949933262nt_int
        @ ( map_in7157766398909135175nt_int
          @ ^ [X: int] : ( product_Pair_int_int @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4033_Id__on__set,axiom,
    ! [Xs2: list_Code_integer] :
      ( ( id_on_Code_integer @ ( set_Code_integer2 @ Xs2 ) )
      = ( set_Pr920681315882439344nteger
        @ ( map_Co3589949550033412536nteger
          @ ^ [X: code_integer] : ( produc1086072967326762835nteger @ X @ X )
          @ Xs2 ) ) ) ).

% Id_on_set
thf(fact_4034_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N )
              & ( distinct_VEBT_VEBT @ Xs )
              & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4035_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ( size_s3451745648224563538omplex @ Xs )
                = N )
              & ( distinct_complex @ Xs )
              & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4036_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_Code_integer,N: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ( size_s3445333598471063425nteger @ Xs )
                = N )
              & ( distin1543349897113766820nteger @ Xs )
              & ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4037_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_real,N: nat] :
      ( ( finite_finite_real @ A3 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ( size_size_list_real @ Xs )
                = N )
              & ( distinct_real @ Xs )
              & ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4038_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ( size_size_list_o @ Xs )
                = N )
              & ( distinct_o @ Xs )
              & ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4039_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = N )
              & ( distinct_nat @ Xs )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4040_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ( size_size_list_int @ Xs )
                = N )
              & ( distinct_int @ Xs )
              & ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_4041_length__product__lists,axiom,
    ! [Xss: list_list_real] :
      ( ( size_s6660260683639930848t_real @ ( product_lists_real @ Xss ) )
      = ( foldr_nat_nat @ times_times_nat @ ( map_list_real_nat @ size_size_list_real @ Xss ) @ one_one_nat ) ) ).

% length_product_lists
thf(fact_4042_length__product__lists,axiom,
    ! [Xss: list_list_o] :
      ( ( size_s2710708370519433104list_o @ ( product_lists_o @ Xss ) )
      = ( foldr_nat_nat @ times_times_nat @ ( map_list_o_nat @ size_size_list_o @ Xss ) @ one_one_nat ) ) ).

% length_product_lists
thf(fact_4043_length__product__lists,axiom,
    ! [Xss: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( product_lists_nat @ Xss ) )
      = ( foldr_nat_nat @ times_times_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xss ) @ one_one_nat ) ) ).

% length_product_lists
thf(fact_4044_length__product__lists,axiom,
    ! [Xss: list_list_int] :
      ( ( size_s533118279054570080st_int @ ( product_lists_int @ Xss ) )
      = ( foldr_nat_nat @ times_times_nat @ ( map_list_int_nat @ size_size_list_int @ Xss ) @ one_one_nat ) ) ).

% length_product_lists
thf(fact_4045_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_4046_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( B
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_4047_Id__on__empty,axiom,
    ( ( id_on_real @ bot_bot_set_real )
    = bot_bo3948376660626123781l_real ) ).

% Id_on_empty
thf(fact_4048_Id__on__empty,axiom,
    ( ( id_on_o @ bot_bot_set_o )
    = bot_bo7073875226086086771od_o_o ) ).

% Id_on_empty
thf(fact_4049_Id__on__empty,axiom,
    ( ( id_on_nat @ bot_bot_set_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Id_on_empty
thf(fact_4050_Id__on__empty,axiom,
    ( ( id_on_int @ bot_bot_set_int )
    = bot_bo1796632182523588997nt_int ) ).

% Id_on_empty
thf(fact_4051_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_4052_ent__iffI,axiom,
    ! [A3: assn,B4: assn] :
      ( ( entails @ A3 @ B4 )
     => ( ( entails @ B4 @ A3 )
       => ( A3 = B4 ) ) ) ).

% ent_iffI
thf(fact_4053_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_4054_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_4055_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_4056_fr__rot,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( entails @ ( times_times_assn @ A3 @ B4 ) @ C2 )
     => ( entails @ ( times_times_assn @ B4 @ A3 ) @ C2 ) ) ).

% fr_rot
thf(fact_4057_fr__refl,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( entails @ A3 @ B4 )
     => ( entails @ ( times_times_assn @ A3 @ C2 ) @ ( times_times_assn @ B4 @ C2 ) ) ) ).

% fr_refl
thf(fact_4058_fr__rot__rhs,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ B4 @ C2 ) )
     => ( entails @ A3 @ ( times_times_assn @ C2 @ B4 ) ) ) ).

% fr_rot_rhs
thf(fact_4059_ent__star__mono,axiom,
    ! [P: assn,P4: assn,Q: assn,Q3: assn] :
      ( ( entails @ P @ P4 )
     => ( ( entails @ Q @ Q3 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P4 @ Q3 ) ) ) ) ).

% ent_star_mono
thf(fact_4060_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F5: assn,Q: assn] :
      ( ( entails @ P @ R )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( ( entails @ ( times_times_assn @ R @ F5 ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_4061_cons__rule,axiom,
    ! [P: assn,P4: assn,Q: option_nat > assn,Q3: option_nat > assn,C: heap_T2636463487746394924on_nat] :
      ( ( entails @ P @ P4 )
     => ( ! [X3: option_nat] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( ( hoare_7629718768684598413on_nat @ P4 @ C @ Q )
         => ( hoare_7629718768684598413on_nat @ P @ C @ Q3 ) ) ) ) ).

% cons_rule
thf(fact_4062_cons__rule,axiom,
    ! [P: assn,P4: assn,Q: $o > assn,Q3: $o > assn,C: heap_Time_Heap_o] :
      ( ( entails @ P @ P4 )
     => ( ! [X3: $o] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( ( hoare_hoare_triple_o @ P4 @ C @ Q )
         => ( hoare_hoare_triple_o @ P @ C @ Q3 ) ) ) ) ).

% cons_rule
thf(fact_4063_cons__rule,axiom,
    ! [P: assn,P4: assn,Q: vEBT_VEBTi > assn,Q3: vEBT_VEBTi > assn,C: heap_T8145700208782473153_VEBTi] :
      ( ( entails @ P @ P4 )
     => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( ( hoare_1429296392585015714_VEBTi @ P4 @ C @ Q )
         => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q3 ) ) ) ) ).

% cons_rule
thf(fact_4064_cons__rule,axiom,
    ! [P: assn,P4: assn,Q: nat > assn,Q3: nat > assn,C: heap_Time_Heap_nat] :
      ( ( entails @ P @ P4 )
     => ( ! [X3: nat] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( ( hoare_3067605981109127869le_nat @ P4 @ C @ Q )
         => ( hoare_3067605981109127869le_nat @ P @ C @ Q3 ) ) ) ) ).

% cons_rule
thf(fact_4065_cons__post__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,Q3: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ! [X3: option_nat] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( hoare_7629718768684598413on_nat @ P @ C @ Q3 ) ) ) ).

% cons_post_rule
thf(fact_4066_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,Q3: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ! [X3: $o] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( hoare_hoare_triple_o @ P @ C @ Q3 ) ) ) ).

% cons_post_rule
thf(fact_4067_cons__post__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Q3: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q3 ) ) ) ).

% cons_post_rule
thf(fact_4068_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,Q3: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ! [X3: nat] : ( entails @ ( Q @ X3 ) @ ( Q3 @ X3 ) )
       => ( hoare_3067605981109127869le_nat @ P @ C @ Q3 ) ) ) ).

% cons_post_rule
thf(fact_4069_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_4070_cons__pre__rule,axiom,
    ! [P: assn,P4: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( entails @ P @ P4 )
     => ( ( hoare_7629718768684598413on_nat @ P4 @ C @ Q )
       => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_4071_cons__pre__rule,axiom,
    ! [P: assn,P4: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ P4 )
     => ( ( hoare_hoare_triple_o @ P4 @ C @ Q )
       => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_4072_cons__pre__rule,axiom,
    ! [P: assn,P4: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ P4 )
     => ( ( hoare_1429296392585015714_VEBTi @ P4 @ C @ Q )
       => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_4073_cons__pre__rule,axiom,
    ! [P: assn,P4: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( entails @ P @ P4 )
     => ( ( hoare_3067605981109127869le_nat @ P4 @ C @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_4074_fi__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,Ps: assn,F5: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( hoare_7629718768684598413on_nat @ Ps @ C
          @ ^ [X: option_nat] : ( times_times_assn @ ( Q @ X ) @ F5 ) ) ) ) ).

% fi_rule
thf(fact_4075_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,Ps: assn,F5: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( hoare_hoare_triple_o @ Ps @ C
          @ ^ [X: $o] : ( times_times_assn @ ( Q @ X ) @ F5 ) ) ) ) ).

% fi_rule
thf(fact_4076_fi__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Ps: assn,F5: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( hoare_1429296392585015714_VEBTi @ Ps @ C
          @ ^ [X: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X ) @ F5 ) ) ) ) ).

% fi_rule
thf(fact_4077_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,Ps: assn,F5: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F5 ) )
       => ( hoare_3067605981109127869le_nat @ Ps @ C
          @ ^ [X: nat] : ( times_times_assn @ ( Q @ X ) @ F5 ) ) ) ) ).

% fi_rule
thf(fact_4078_htt__cons__rule,axiom,
    ! [P4: assn,C: heap_T2636463487746394924on_nat,Q3: option_nat > assn,T4: nat,P: assn,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ P4 @ C @ Q3 @ T4 )
     => ( ( entails @ P @ P4 )
       => ( ! [X3: option_nat] : ( entails @ ( Q3 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T4 @ T )
           => ( time_htt_option_nat @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_4079_htt__cons__rule,axiom,
    ! [P4: assn,C: heap_T8145700208782473153_VEBTi,Q3: vEBT_VEBTi > assn,T4: nat,P: assn,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P4 @ C @ Q3 @ T4 )
     => ( ( entails @ P @ P4 )
       => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q3 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T4 @ T )
           => ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_4080_htt__cons__rule,axiom,
    ! [P4: assn,C: heap_Time_Heap_o,Q3: $o > assn,T4: nat,P: assn,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ P4 @ C @ Q3 @ T4 )
     => ( ( entails @ P @ P4 )
       => ( ! [X3: $o] : ( entails @ ( Q3 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T4 @ T )
           => ( time_htt_o @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_4081_htt__cons__rule,axiom,
    ! [P4: assn,C: heap_Time_Heap_nat,Q3: nat > assn,T4: nat,P: assn,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ P4 @ C @ Q3 @ T4 )
     => ( ( entails @ P @ P4 )
       => ( ! [X3: nat] : ( entails @ ( Q3 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T4 @ T )
           => ( time_htt_nat @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_4082_distinct__finite__set,axiom,
    ! [X4: set_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT
      @ ( collec5608196760682091941T_VEBT
        @ ^ [Ys3: list_VEBT_VEBT] :
            ( ( ( set_VEBT_VEBT2 @ Ys3 )
              = X4 )
            & ( distinct_VEBT_VEBT @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_4083_distinct__finite__set,axiom,
    ! [X4: set_nat] :
      ( finite8100373058378681591st_nat
      @ ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( set_nat2 @ Ys3 )
              = X4 )
            & ( distinct_nat @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_4084_distinct__finite__set,axiom,
    ! [X4: set_real] :
      ( finite306553202115118035t_real
      @ ( collect_list_real
        @ ^ [Ys3: list_real] :
            ( ( ( set_real2 @ Ys3 )
              = X4 )
            & ( distinct_real @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_4085_distinct__finite__set,axiom,
    ! [X4: set_o] :
      ( finite_finite_list_o
      @ ( collect_list_o
        @ ^ [Ys3: list_o] :
            ( ( ( set_o2 @ Ys3 )
              = X4 )
            & ( distinct_o @ Ys3 ) ) ) ) ).

% distinct_finite_set
thf(fact_4086_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_4087_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_4088_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_4089_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_4090_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_4091_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBTi
    = ( ^ [Xs: list_VEBT_VEBTi] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                 != ( nth_VEBT_VEBTi @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4092_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBT
    = ( ^ [Xs: list_VEBT_VEBT] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                 != ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4093_distinct__conv__nth,axiom,
    ( distinct_real
    = ( ^ [Xs: list_real] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_real @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_real @ Xs @ I3 )
                 != ( nth_real @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4094_distinct__conv__nth,axiom,
    ( distinct_o
    = ( ^ [Xs: list_o] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_o @ Xs @ I3 )
                 != ( nth_o @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4095_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs: list_nat] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_nat @ Xs @ I3 )
                 != ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4096_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs: list_int] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ( I3 != J3 )
               => ( ( nth_int @ Xs @ I3 )
                 != ( nth_int @ Xs @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_4097_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBTi @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( ( ( nth_VEBT_VEBTi @ Xs2 @ I )
              = ( nth_VEBT_VEBTi @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4098_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( ( ( nth_VEBT_VEBT @ Xs2 @ I )
              = ( nth_VEBT_VEBT @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4099_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_real,I: nat,J: nat] :
      ( ( distinct_real @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
         => ( ( ( nth_real @ Xs2 @ I )
              = ( nth_real @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4100_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_o,I: nat,J: nat] :
      ( ( distinct_o @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
         => ( ( ( nth_o @ Xs2 @ I )
              = ( nth_o @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4101_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
         => ( ( ( nth_nat @ Xs2 @ I )
              = ( nth_nat @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4102_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_int,I: nat,J: nat] :
      ( ( distinct_int @ Xs2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
         => ( ( ( nth_int @ Xs2 @ I )
              = ( nth_int @ Xs2 @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_4103_distinct__length__le,axiom,
    ! [Ys: list_VEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Ys )
     => ( ( ( set_VEBT_VEBT2 @ Ys )
          = ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).

% distinct_length_le
thf(fact_4104_distinct__length__le,axiom,
    ! [Ys: list_real,Xs2: list_real] :
      ( ( distinct_real @ Ys )
     => ( ( ( set_real2 @ Ys )
          = ( set_real2 @ Xs2 ) )
       => ( ord_less_eq_nat @ ( size_size_list_real @ Ys ) @ ( size_size_list_real @ Xs2 ) ) ) ) ).

% distinct_length_le
thf(fact_4105_distinct__length__le,axiom,
    ! [Ys: list_o,Xs2: list_o] :
      ( ( distinct_o @ Ys )
     => ( ( ( set_o2 @ Ys )
          = ( set_o2 @ Xs2 ) )
       => ( ord_less_eq_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) ) ) ) ).

% distinct_length_le
thf(fact_4106_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs2: list_nat] :
      ( ( distinct_nat @ Ys )
     => ( ( ( set_nat2 @ Ys )
          = ( set_nat2 @ Xs2 ) )
       => ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% distinct_length_le
thf(fact_4107_distinct__length__le,axiom,
    ! [Ys: list_int,Xs2: list_int] :
      ( ( distinct_int @ Ys )
     => ( ( ( set_int2 @ Ys )
          = ( set_int2 @ Xs2 ) )
       => ( ord_less_eq_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) ) ) ) ).

% distinct_length_le
thf(fact_4108_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
     => ( ( distinct_real @ Xs2 )
       => ( ( sorted_wrt_real @ ord_less_eq_real @ Ys )
         => ( ( distinct_real @ Ys )
           => ( ( ( set_real2 @ Xs2 )
                = ( set_real2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4109_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( ( distinct_o @ Xs2 )
       => ( ( sorted_wrt_o @ ord_less_eq_o @ Ys )
         => ( ( distinct_o @ Ys )
           => ( ( ( set_o2 @ Xs2 )
                = ( set_o2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4110_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( ( distinct_rat @ Xs2 )
       => ( ( sorted_wrt_rat @ ord_less_eq_rat @ Ys )
         => ( ( distinct_rat @ Ys )
           => ( ( ( set_rat2 @ Xs2 )
                = ( set_rat2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4111_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( ( distinct_num @ Xs2 )
       => ( ( sorted_wrt_num @ ord_less_eq_num @ Ys )
         => ( ( distinct_num @ Ys )
           => ( ( ( set_num2 @ Xs2 )
                = ( set_num2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4112_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( ( distinct_nat @ Xs2 )
       => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
         => ( ( distinct_nat @ Ys )
           => ( ( ( set_nat2 @ Xs2 )
                = ( set_nat2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4113_sorted__distinct__set__unique,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( ( distinct_int @ Xs2 )
       => ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
         => ( ( distinct_int @ Ys )
           => ( ( ( set_int2 @ Xs2 )
                = ( set_int2 @ Ys ) )
             => ( Xs2 = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_4114_finite__sorted__distinct__unique,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ? [X3: list_real] :
          ( ( ( set_real2 @ X3 )
            = A3 )
          & ( sorted_wrt_real @ ord_less_eq_real @ X3 )
          & ( distinct_real @ X3 )
          & ! [Y5: list_real] :
              ( ( ( ( set_real2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_real @ ord_less_eq_real @ Y5 )
                & ( distinct_real @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4115_finite__sorted__distinct__unique,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ? [X3: list_o] :
          ( ( ( set_o2 @ X3 )
            = A3 )
          & ( sorted_wrt_o @ ord_less_eq_o @ X3 )
          & ( distinct_o @ X3 )
          & ! [Y5: list_o] :
              ( ( ( ( set_o2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_o @ ord_less_eq_o @ Y5 )
                & ( distinct_o @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4116_finite__sorted__distinct__unique,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ? [X3: list_Code_integer] :
          ( ( ( set_Code_integer2 @ X3 )
            = A3 )
          & ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ X3 )
          & ( distin1543349897113766820nteger @ X3 )
          & ! [Y5: list_Code_integer] :
              ( ( ( ( set_Code_integer2 @ Y5 )
                  = A3 )
                & ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ Y5 )
                & ( distin1543349897113766820nteger @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4117_finite__sorted__distinct__unique,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ? [X3: list_rat] :
          ( ( ( set_rat2 @ X3 )
            = A3 )
          & ( sorted_wrt_rat @ ord_less_eq_rat @ X3 )
          & ( distinct_rat @ X3 )
          & ! [Y5: list_rat] :
              ( ( ( ( set_rat2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_rat @ ord_less_eq_rat @ Y5 )
                & ( distinct_rat @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4118_finite__sorted__distinct__unique,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ? [X3: list_num] :
          ( ( ( set_num2 @ X3 )
            = A3 )
          & ( sorted_wrt_num @ ord_less_eq_num @ X3 )
          & ( distinct_num @ X3 )
          & ! [Y5: list_num] :
              ( ( ( ( set_num2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_num @ ord_less_eq_num @ Y5 )
                & ( distinct_num @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4119_finite__sorted__distinct__unique,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [X3: list_nat] :
          ( ( ( set_nat2 @ X3 )
            = A3 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
          & ( distinct_nat @ X3 )
          & ! [Y5: list_nat] :
              ( ( ( ( set_nat2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
                & ( distinct_nat @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4120_finite__sorted__distinct__unique,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ? [X3: list_int] :
          ( ( ( set_int2 @ X3 )
            = A3 )
          & ( sorted_wrt_int @ ord_less_eq_int @ X3 )
          & ( distinct_int @ X3 )
          & ! [Y5: list_int] :
              ( ( ( ( set_int2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_int @ ord_less_eq_int @ Y5 )
                & ( distinct_int @ Y5 ) )
             => ( Y5 = X3 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_4121_distinct__Ex1,axiom,
    ! [Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( distinct_VEBT_VEBTi @ Xs2 )
     => ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
                  & ( ( nth_VEBT_VEBTi @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4122_distinct__Ex1,axiom,
    ! [Xs2: list_set_nat,X4: set_nat] :
      ( ( distinct_set_nat @ Xs2 )
     => ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
                  & ( ( nth_set_nat @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4123_distinct__Ex1,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                  & ( ( nth_VEBT_VEBT @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4124_distinct__Ex1,axiom,
    ! [Xs2: list_real,X4: real] :
      ( ( distinct_real @ Xs2 )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_real @ Xs2 ) )
                  & ( ( nth_real @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4125_distinct__Ex1,axiom,
    ! [Xs2: list_o,X4: $o] :
      ( ( distinct_o @ Xs2 )
     => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_o @ Xs2 ) )
                  & ( ( nth_o @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4126_distinct__Ex1,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs2 ) )
                  & ( ( nth_nat @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4127_distinct__Ex1,axiom,
    ! [Xs2: list_int,X4: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ X3 )
              = X4 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_int @ Xs2 ) )
                  & ( ( nth_int @ Xs2 @ Y5 )
                    = X4 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_4128_distinct__idx,axiom,
    ! [F: vEBT_VEBT > nat,L: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_nat @ ( map_VEBT_VEBT_nat @ F @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
              = ( F @ ( nth_VEBT_VEBT @ L @ J ) ) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
thf(fact_4129_distinct__idx,axiom,
    ! [F: vEBT_VEBT > real,L: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_real @ ( map_VEBT_VEBT_real @ F @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
              = ( F @ ( nth_VEBT_VEBT @ L @ J ) ) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
thf(fact_4130_distinct__idx,axiom,
    ! [F: product_prod_o_o > $o,L: list_P4002435161011370285od_o_o,I: nat,J: nat] :
      ( ( distinct_o @ ( map_Pr7541730621154948341_o_o_o @ F @ L ) )
     => ( ( ord_less_nat @ I @ ( size_s1515746228057227161od_o_o @ L ) )
       => ( ( ord_less_nat @ J @ ( size_s1515746228057227161od_o_o @ L ) )
         => ( ( ( F @ ( nth_Product_prod_o_o @ L @ I ) )
              = ( F @ ( nth_Product_prod_o_o @ L @ J ) ) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
thf(fact_4131_distinct__idx,axiom,
    ! [F: nat > nat,L: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ ( map_nat_nat @ F @ L ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
         => ( ( ( F @ ( nth_nat @ L @ I ) )
              = ( F @ ( nth_nat @ L @ J ) ) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
thf(fact_4132_distinct__idx,axiom,
    ! [F: nat > $o,L: list_nat,I: nat,J: nat] :
      ( ( distinct_o @ ( map_nat_o @ F @ L ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
         => ( ( ( F @ ( nth_nat @ L @ I ) )
              = ( F @ ( nth_nat @ L @ J ) ) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
thf(fact_4133_distinct__finite__subset,axiom,
    ! [X4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ X4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Ys3: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Ys3 ) @ X4 )
              & ( distinct_VEBT_VEBT @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4134_distinct__finite__subset,axiom,
    ! [X4: set_real] :
      ( ( finite_finite_real @ X4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Ys3: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Ys3 ) @ X4 )
              & ( distinct_real @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4135_distinct__finite__subset,axiom,
    ! [X4: set_o] :
      ( ( finite_finite_o @ X4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Ys3: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Ys3 ) @ X4 )
              & ( distinct_o @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4136_distinct__finite__subset,axiom,
    ! [X4: set_nat] :
      ( ( finite_finite_nat @ X4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Ys3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ X4 )
              & ( distinct_nat @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4137_distinct__finite__subset,axiom,
    ! [X4: set_complex] :
      ( ( finite3207457112153483333omplex @ X4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Ys3: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Ys3 ) @ X4 )
              & ( distinct_complex @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4138_distinct__finite__subset,axiom,
    ! [X4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ X4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Ys3: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Ys3 ) @ X4 )
              & ( distin1543349897113766820nteger @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4139_distinct__finite__subset,axiom,
    ! [X4: set_int] :
      ( ( finite_finite_int @ X4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Ys3: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Ys3 ) @ X4 )
              & ( distinct_int @ Ys3 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_4140_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_4141_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_4142_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_4143_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_4144_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_4145_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_4146_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_4147_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_4148_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_4149_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_4150_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_4151_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_4152_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_4153_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_4154_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_4155_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_4156_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_4157_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_4158_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_4159_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBTi > nat > assn,Xs2: list_VEBT_VEBTi,Xsi: list_nat,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4160_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4161_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBT > assn,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4162_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBTi > int > assn,Xs2: list_VEBT_VEBTi,Xsi: list_int,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4163_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,Xsi: list_nat,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4164_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4165_rule__at__index,axiom,
    ! [P: assn,A3: vEBT_VEBT > int > assn,Xs2: list_VEBT_VEBT,Xsi: list_int,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8294436054247626077BT_int @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L8294436054247626077BT_int @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4166_rule__at__index,axiom,
    ! [P: assn,A3: real > nat > assn,Xs2: list_real,Xsi: list_nat,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4167_rule__at__index,axiom,
    ! [P: assn,A3: real > vEBT_VEBTi > assn,Xs2: list_real,Xsi: list_VEBT_VEBTi,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4168_rule__at__index,axiom,
    ! [P: assn,A3: real > vEBT_VEBT > assn,Xs2: list_real,Xsi: list_VEBT_VEBT,F5: assn,I: nat,C: heap_Time_Heap_o,Q3: $o > assn,F7: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C @ Q3 )
         => ( ! [R4: $o] : ( entails @ ( Q3 @ R4 ) @ ( times_times_assn @ ( times_times_assn @ ( A3 @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F7 @ R4 ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R2: $o] : ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A3 @ Xs2 @ Xsi ) @ ( F7 @ R2 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_4169_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_4170_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_4171_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_4172_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_4173_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_4174_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_4175_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_4176_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_4177_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_4178_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_4179_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBTi > assn,X4: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4180_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBT > assn,X4: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4181_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: vEBT_VEBT > vEBT_VEBT > assn,X4: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4182_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: real > vEBT_VEBTi > assn,X4: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4183_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: real > vEBT_VEBT > assn,X4: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBT,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4184_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: $o > vEBT_VEBTi > assn,X4: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4185_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: $o > vEBT_VEBT > assn,X4: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBT,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4186_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: nat > vEBT_VEBTi > assn,X4: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4187_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: nat > vEBT_VEBT > assn,X4: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBT,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4188_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A3: int > vEBT_VEBTi > assn,X4: int,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_int,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A3 @ ( list_update_int @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_4189_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X4: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_list_update
thf(fact_4190_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X4: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_4191_length__list__update,axiom,
    ! [Xs2: list_real,I: nat,X4: real] :
      ( ( size_size_list_real @ ( list_update_real @ Xs2 @ I @ X4 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_list_update
thf(fact_4192_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X4: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X4 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_4193_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X4: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X4 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_4194_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X4: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X4 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_4195_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_4196_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_4197_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_4198_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_4199_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_4200_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_4201_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_4202_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_4203_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
thf(fact_4204_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_4205_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_4206_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_4207_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_4208_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_4209_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_4210_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_4211_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_4212_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_4213_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_4214_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_4215_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_4216_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_4217_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_4218_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_4219_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_4220_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_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_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_4231_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_4232_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_4233_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_4234_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_4235_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_4236_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_4237_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X4: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4238_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4239_list__update__beyond,axiom,
    ! [Xs2: list_real,I: nat,X4: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( list_update_real @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4240_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X4: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4241_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4242_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X4: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X4 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4243_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_4244_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_4245_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_4246_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F5: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C @ Q )
           => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_4247_snga__same__false,axiom,
    ! [P5: array_VEBT_VEBTi,X4: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ P5 @ X4 ) @ ( snga_assn_VEBT_VEBTi @ P5 @ Y ) )
      = bot_bot_assn ) ).

% snga_same_false
thf(fact_4248_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_real,Y: vEBT_VEBT,X4: 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 @ X4 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) ) @ ( 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 @ X4 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ) ).

% tcd
thf(fact_4249_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_o,Y: vEBT_VEBT,X4: 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 @ X4 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) ) @ ( 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 @ X4 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ) ).

% tcd
thf(fact_4250_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_nat,Y: vEBT_VEBT,X4: 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 @ X4 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) ) @ ( 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 @ X4 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ) ).

% tcd
thf(fact_4251_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_int,Y: vEBT_VEBT,X4: 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_int @ TreeList3 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X4 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) ) @ ( 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 @ X4 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ) ).

% tcd
thf(fact_4252_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4253_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4254_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_real,X4: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4255_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X4: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4256_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4257_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X4: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X4 ) @ I )
        = X4 ) ) ).

% nth_list_update_eq
thf(fact_4258_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,X4: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ J @ X4 ) @ I )
        = ( nth_VEBT_VEBTi @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4259_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,X4: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ J @ X4 ) @ I )
        = ( nth_VEBT_VEBT @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4260_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,X4: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( nth_real @ ( list_update_real @ L @ J @ X4 ) @ I )
        = ( nth_real @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4261_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,X4: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( nth_o @ ( list_update_o @ L @ J @ X4 ) @ I )
        = ( nth_o @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4262_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X4: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( list_update_nat @ L @ J @ X4 ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4263_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X4: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( list_update_int @ L @ J @ X4 ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4264_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_4265_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X4: 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 @ X4 )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X4 ) ) @ ( 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 @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_4266_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L: nat,X4: 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 ) @ X4 )
            @ ( 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 @ X4 ) ) @ ( 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 @ X4 ) ) @ ( 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 @ X4 ) ) ) ) ) ).

% big_assn_simp
thf(fact_4267_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4268_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4269_set__swap,axiom,
    ! [I: nat,Xs2: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs2 @ I @ ( nth_real @ Xs2 @ J ) ) @ J @ ( nth_real @ Xs2 @ I ) ) )
          = ( set_real2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4270_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4271_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4272_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4273_distinct__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( distinct_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs2 @ I ) ) )
          = ( distinct_VEBT_VEBTi @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4274_distinct__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( distinct_VEBT_VEBT @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4275_distinct__swap,axiom,
    ! [I: nat,Xs2: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
       => ( ( distinct_real @ ( list_update_real @ ( list_update_real @ Xs2 @ I @ ( nth_real @ Xs2 @ J ) ) @ J @ ( nth_real @ Xs2 @ I ) ) )
          = ( distinct_real @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4276_distinct__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( distinct_o @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( distinct_o @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4277_distinct__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( distinct_nat @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4278_distinct__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( distinct_int @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_4279_snga__prec,axiom,
    ( precis1518117471652397933_VEBTi
    @ ^ [X: list_VEBT_VEBTi,P6: array_VEBT_VEBTi] : ( snga_assn_VEBT_VEBTi @ P6 @ X ) ) ).

% snga_prec
thf(fact_4280_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4281_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4282_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4283_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4284_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4285_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4286_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4287_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4288_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4289_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4290_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4291_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4292_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ( set_or66887138388493659n_real @ A @ B )
            = ( set_or66887138388493659n_real @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4293_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4294_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C @ D )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4295_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4296_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4297_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4298_set__update__subsetI,axiom,
    ! [Xs2: list_set_nat,A3: set_set_nat,X4: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A3 )
     => ( ( member_set_nat @ X4 @ A3 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4299_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A3: set_nat,X4: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
     => ( ( member_nat @ X4 @ A3 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4300_set__update__subsetI,axiom,
    ! [Xs2: list_real,A3: set_real,X4: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A3 )
     => ( ( member_real @ X4 @ A3 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4301_set__update__subsetI,axiom,
    ! [Xs2: list_o,A3: set_o,X4: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
     => ( ( member_o @ X4 @ A3 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4302_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBTi,A3: set_VEBT_VEBTi,X4: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ A3 )
     => ( ( member_VEBT_VEBTi @ X4 @ A3 )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4303_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A3: set_VEBT_VEBT,X4: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
     => ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4304_set__update__subsetI,axiom,
    ! [Xs2: list_int,A3: set_int,X4: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
     => ( ( member_int @ X4 @ A3 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4305_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_4306_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_4307_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ B @ A )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4308_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4309_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4310_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4311_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4312_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_4313_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_4314_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_4315_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_4316_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_set_nat,X4: set_nat,Y: set_nat] :
      ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
     => ( ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: set_nat] : ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4317_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X4: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4318_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X4: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4319_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X4: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: real] : ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4320_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,X4: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: $o] : ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4321_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: nat] : ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4322_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X4: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ! [Y4: int] : ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ L @ I @ Y4 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4323_in__set__upd__cases,axiom,
    ! [X4: set_nat,L: list_set_nat,I: nat,Y: set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
         => ( X4 != Y ) )
       => ( member_set_nat @ X4 @ ( set_set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4324_in__set__upd__cases,axiom,
    ! [X4: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
         => ( X4 != Y ) )
       => ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4325_in__set__upd__cases,axiom,
    ! [X4: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( X4 != Y ) )
       => ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4326_in__set__upd__cases,axiom,
    ! [X4: real,L: list_real,I: nat,Y: real] :
      ( ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( X4 != Y ) )
       => ( member_real @ X4 @ ( set_real2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4327_in__set__upd__cases,axiom,
    ! [X4: $o,L: list_o,I: nat,Y: $o] :
      ( ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( X4 = ~ Y ) )
       => ( member_o @ X4 @ ( set_o2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4328_in__set__upd__cases,axiom,
    ! [X4: nat,L: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( X4 != Y ) )
       => ( member_nat @ X4 @ ( set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4329_in__set__upd__cases,axiom,
    ! [X4: int,L: list_int,I: nat,Y: int] :
      ( ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( X4 != Y ) )
       => ( member_int @ X4 @ ( set_int2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4330_in__set__upd__eq,axiom,
    ! [I: nat,L: list_set_nat,X4: set_nat,Y: set_nat] :
      ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ L ) )
     => ( ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_set_nat @ X4 @ ( set_set_nat2 @ L ) )
            & ! [Y4: set_nat] : ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4331_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X4: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ L ) )
            & ! [Y4: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4332_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X4: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ L ) )
            & ! [Y4: vEBT_VEBT] : ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4333_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X4: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_real @ X4 @ ( set_real2 @ L ) )
            & ! [Y4: real] : ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4334_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,X4: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_o @ X4 @ ( set_o2 @ L ) )
            & ! [Y4: $o] : ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4335_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_nat @ X4 @ ( set_nat2 @ L ) )
            & ! [Y4: nat] : ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4336_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X4: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X4 = Y )
          | ( ( member_int @ X4 @ ( set_int2 @ L ) )
            & ! [Y4: int] : ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ L @ I @ Y4 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4337_set__update__memI,axiom,
    ! [N: nat,Xs2: list_set_nat,X4: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ X4 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4338_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4339_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4340_set__update__memI,axiom,
    ! [N: nat,Xs2: list_real,X4: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X4 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4341_set__update__memI,axiom,
    ! [N: nat,Xs2: list_o,X4: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X4 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4342_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X4: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X4 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4343_set__update__memI,axiom,
    ! [N: nat,Xs2: list_int,X4: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X4 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_4344_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X4: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X4 ) ) @ ( insert_nat @ X4 @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4345_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I: nat,X4: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X4 ) ) @ ( insert_real @ X4 @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4346_set__update__subset__insert,axiom,
    ! [Xs2: list_o,I: nat,X4: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I @ X4 ) ) @ ( insert_o @ X4 @ ( set_o2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4347_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X4: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) ) @ ( insert_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4348_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X4: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) ) @ ( insert_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4349_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X4: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X4 ) ) @ ( insert_int @ X4 @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4350_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_VEBT_VEBTi @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4351_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4352_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_real,X4: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( list_update_real @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_real @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4353_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X4: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4354_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4355_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X4: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X4 )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X4 ) ) ) ).

% list_update_same_conv
thf(fact_4356_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ J )
            = X4 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ J )
            = ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4357_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) @ J )
            = X4 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4358_nth__list__update,axiom,
    ! [I: nat,Xs2: list_real,J: nat,X4: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X4 ) @ J )
            = X4 ) )
        & ( ( I != J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X4 ) @ J )
            = ( nth_real @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4359_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X4: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X4 ) @ J )
        = ( ( ( I = J )
           => X4 )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4360_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X4 ) @ J )
            = X4 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X4 ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4361_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X4: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X4 ) @ J )
            = X4 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X4 ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4362_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X4 ) @ J )
          = X4 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X4 ) @ J )
          = ( nth_VEBT_VEBTi @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4363_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X4 ) @ J )
          = X4 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X4 ) @ J )
          = ( nth_VEBT_VEBT @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4364_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_real,X4: real] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X4 ) @ J )
          = X4 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X4 ) @ J )
          = ( nth_real @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4365_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,X4: $o,J: nat] :
      ( ( nth_o @ ( list_update_o @ L @ I @ X4 ) @ J )
      = ( ( ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => X4 )
        & ( ~ ( ( I = J )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => ( nth_o @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4366_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X4: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X4 ) @ J )
          = X4 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X4 ) @ J )
          = ( nth_nat @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4367_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X4: int] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X4 ) @ J )
          = X4 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X4 ) @ J )
          = ( nth_int @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_4368_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4369_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4370_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4371_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4372_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4373_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_o,A3: $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 @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L6286945158656146733_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4374_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_o,A3: $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 @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L1319876754960170684T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4375_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_nat,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L7489483478785760935_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4376_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_nat,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( vEBT_L8511957252848910786T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4377_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_int,A3: int > vEBT_VEBTi > assn,X1: int,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
       => ( ( vEBT_L114188773329725699_VEBTi @ I5 @ A3 @ ( list_update_int @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( vEBT_L114188773329725699_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4378_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_4379_map__upd__eq,axiom,
    ! [I: nat,L: list_P4002435161011370285od_o_o,F: product_prod_o_o > $o,X4: product_prod_o_o] :
      ( ( ( ord_less_nat @ I @ ( size_s1515746228057227161od_o_o @ L ) )
       => ( ( F @ ( nth_Product_prod_o_o @ L @ I ) )
          = ( F @ X4 ) ) )
     => ( ( map_Pr7541730621154948341_o_o_o @ F @ ( list_u1537252308907898773od_o_o @ L @ I @ X4 ) )
        = ( map_Pr7541730621154948341_o_o_o @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_4380_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > nat,X4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X4 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X4 ) )
        = ( map_VEBT_VEBT_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_4381_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > real,X4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X4 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X4 ) )
        = ( map_VEBT_VEBT_real @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_4382_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > nat,X4: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X4 ) ) )
     => ( ( map_nat_nat @ F @ ( list_update_nat @ L @ I @ X4 ) )
        = ( map_nat_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_4383_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > $o,X4: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X4 ) ) )
     => ( ( map_nat_o @ F @ ( list_update_nat @ L @ I @ X4 ) )
        = ( map_nat_o @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_4384_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_4385_listI__assn__conv,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A3 @ Xs2 @ Xsi )
        = ( vEBT_L6296928887356842470_VEBTi @ A3 @ Xs2 @ Xsi ) ) ) ).

% listI_assn_conv
thf(fact_4386_list__assn__conv__idx,axiom,
    ( vEBT_L6296928887356842470_VEBTi
    = ( ^ [A5: vEBT_VEBT > vEBT_VEBTi > assn,Xs: list_VEBT_VEBT] : ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ A5 @ Xs ) ) ) ).

% list_assn_conv_idx
thf(fact_4387_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X4: 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 @ X4 ) ) )
        = ( insert_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4388_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X4: 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 @ X4 ) ) )
        = ( insert_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4389_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X4: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( insert_real @ ( nth_real @ L @ I ) @ ( set_real2 @ ( list_update_real @ L @ I @ X4 ) ) )
        = ( insert_real @ X4 @ ( set_real2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4390_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,X4: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( insert_o @ ( nth_o @ L @ I ) @ ( set_o2 @ ( list_update_o @ L @ I @ X4 ) ) )
        = ( insert_o @ X4 @ ( set_o2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4391_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( insert_nat @ ( nth_nat @ L @ I ) @ ( set_nat2 @ ( list_update_nat @ L @ I @ X4 ) ) )
        = ( insert_nat @ X4 @ ( set_nat2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4392_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X4: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( insert_int @ ( nth_int @ L @ I ) @ ( set_int2 @ ( list_update_int @ L @ I @ X4 ) ) )
        = ( insert_int @ X4 @ ( set_int2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4393_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4394_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBTi,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4395_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_VEBT_VEBT,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4396_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4397_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_real,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4398_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_o,A3: $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 @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4399_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_o,A3: $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 @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4400_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_nat,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4401_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_nat,A3: 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 @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4402_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs2: list_int,A3: int > vEBT_VEBTi > assn,X1: int,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
       => ( ( vEBT_L114188773329725699_VEBTi @ ( insert_nat @ I @ I5 ) @ A3 @ ( list_update_int @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X2 ) )
          = ( times_times_assn @ ( A3 @ X1 @ X2 ) @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_4403_listI__assn__conv_H,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,A3: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi,F5: assn] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A3 @ Xs2 @ Xsi ) @ F5 )
        = ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ A3 @ Xs2 @ Xsi ) @ F5 ) ) ) ).

% listI_assn_conv'
thf(fact_4404_distinct__list__update,axiom,
    ! [Xs2: list_set_nat,A: set_nat,I: nat] :
      ( ( distinct_set_nat @ Xs2 )
     => ( ~ ( member_set_nat @ A @ ( minus_2163939370556025621et_nat @ ( set_set_nat2 @ Xs2 ) @ ( insert_set_nat @ ( nth_set_nat @ Xs2 @ I ) @ bot_bot_set_set_nat ) ) )
       => ( distinct_set_nat @ ( list_update_set_nat @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4405_distinct__list__update,axiom,
    ! [Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi,I: nat] :
      ( ( distinct_VEBT_VEBTi @ Xs2 )
     => ( ~ ( member_VEBT_VEBTi @ A @ ( minus_3697805406911847364_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ bot_bo8982466882572371071_VEBTi ) ) )
       => ( distinct_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4406_distinct__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,A: vEBT_VEBT,I: nat] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ~ ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ bot_bo8194388402131092736T_VEBT ) ) )
       => ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4407_distinct__list__update,axiom,
    ! [Xs2: list_real,A: real,I: nat] :
      ( ( distinct_real @ Xs2 )
     => ( ~ ( member_real @ A @ ( minus_minus_set_real @ ( set_real2 @ Xs2 ) @ ( insert_real @ ( nth_real @ Xs2 @ I ) @ bot_bot_set_real ) ) )
       => ( distinct_real @ ( list_update_real @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4408_distinct__list__update,axiom,
    ! [Xs2: list_o,A: $o,I: nat] :
      ( ( distinct_o @ Xs2 )
     => ( ~ ( member_o @ A @ ( minus_minus_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ ( nth_o @ Xs2 @ I ) @ bot_bot_set_o ) ) )
       => ( distinct_o @ ( list_update_o @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4409_distinct__list__update,axiom,
    ! [Xs2: list_int,A: int,I: nat] :
      ( ( distinct_int @ Xs2 )
     => ( ~ ( member_int @ A @ ( minus_minus_set_int @ ( set_int2 @ Xs2 ) @ ( insert_int @ ( nth_int @ Xs2 @ I ) @ bot_bot_set_int ) ) )
       => ( distinct_int @ ( list_update_int @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4410_distinct__list__update,axiom,
    ! [Xs2: list_nat,A: nat,I: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ ( nth_nat @ Xs2 @ I ) @ bot_bot_set_nat ) ) )
       => ( distinct_nat @ ( list_update_nat @ Xs2 @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_4411_set__update__distinct,axiom,
    ! [Xs2: list_VEBT_VEBTi,N: nat,X4: vEBT_VEBTi] :
      ( ( distinct_VEBT_VEBTi @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N @ X4 ) )
          = ( insert_VEBT_VEBTi @ X4 @ ( minus_3697805406911847364_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N ) @ bot_bo8982466882572371071_VEBTi ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4412_set__update__distinct,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X4: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X4 ) )
          = ( insert_VEBT_VEBT @ X4 @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4413_set__update__distinct,axiom,
    ! [Xs2: list_real,N: nat,X4: real] :
      ( ( distinct_real @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ Xs2 @ N @ X4 ) )
          = ( insert_real @ X4 @ ( minus_minus_set_real @ ( set_real2 @ Xs2 ) @ ( insert_real @ ( nth_real @ Xs2 @ N ) @ bot_bot_set_real ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4414_set__update__distinct,axiom,
    ! [Xs2: list_o,N: nat,X4: $o] :
      ( ( distinct_o @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ Xs2 @ N @ X4 ) )
          = ( insert_o @ X4 @ ( minus_minus_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ ( nth_o @ Xs2 @ N ) @ bot_bot_set_o ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4415_set__update__distinct,axiom,
    ! [Xs2: list_int,N: nat,X4: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ Xs2 @ N @ X4 ) )
          = ( insert_int @ X4 @ ( minus_minus_set_int @ ( set_int2 @ Xs2 ) @ ( insert_int @ ( nth_int @ Xs2 @ N ) @ bot_bot_set_int ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4416_set__update__distinct,axiom,
    ! [Xs2: list_nat,N: nat,X4: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X4 ) )
          = ( insert_nat @ X4 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ ( nth_nat @ Xs2 @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_4417_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBTi > assn,X4: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4418_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: vEBT_VEBTi > vEBT_VEBT > assn,X4: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A3 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4419_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: vEBT_VEBT > vEBT_VEBT > assn,X4: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A3 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4420_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: real > vEBT_VEBTi > assn,X4: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBTi,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4421_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: real > vEBT_VEBT > assn,X4: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBT,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A3 @ ( list_update_real @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4422_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: $o > vEBT_VEBTi > assn,X4: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBTi,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4423_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: $o > vEBT_VEBT > assn,X4: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBT,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A3 @ ( list_update_o @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4424_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: nat > vEBT_VEBTi > assn,X4: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBTi,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4425_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: nat > vEBT_VEBT > assn,X4: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBT,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A3 @ ( list_update_nat @ Xs2 @ I @ X4 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4426_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A3: int > vEBT_VEBTi > assn,X4: int,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs2: list_int,Xsi: list_VEBT_VEBTi,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A3 @ X4 @ Xi ) @ ( vEBT_L114188773329725699_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L114188773329725699_VEBTi @ I5 @ A3 @ ( list_update_int @ Xs2 @ I @ X4 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_4427_nth__rule,axiom,
    ! [I: nat,Xs2: list_int,A: array_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( hoare_3065115510600077593le_int @ ( snga_assn_int @ A @ Xs2 ) @ ( array_nth_int @ A @ I )
        @ ^ [R2: int] :
            ( times_times_assn @ ( snga_assn_int @ A @ Xs2 )
            @ ( pure_assn
              @ ( R2
                = ( nth_int @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_4428_nth__rule,axiom,
    ! [I: nat,Xs2: list_option_nat,A: array_option_nat] :
      ( ( ord_less_nat @ I @ ( size_s6086282163384603972on_nat @ Xs2 ) )
     => ( hoare_7629718768684598413on_nat @ ( snga_assn_option_nat @ A @ Xs2 ) @ ( array_nth_option_nat @ A @ I )
        @ ^ [R2: option_nat] :
            ( times_times_assn @ ( snga_assn_option_nat @ A @ Xs2 )
            @ ( pure_assn
              @ ( R2
                = ( nth_option_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_4429_nth__rule,axiom,
    ! [I: nat,Xs2: list_o,A: array_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( hoare_hoare_triple_o @ ( snga_assn_o @ A @ Xs2 ) @ ( array_nth_o @ A @ I )
        @ ^ [R2: $o] :
            ( times_times_assn @ ( snga_assn_o @ A @ Xs2 )
            @ ( pure_assn
              @ ( R2
                = ( nth_o @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_4430_nth__rule,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( hoare_1429296392585015714_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_nth_VEBT_VEBTi @ A @ I )
        @ ^ [R2: vEBT_VEBTi] :
            ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 )
            @ ( pure_assn
              @ ( R2
                = ( nth_VEBT_VEBTi @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_4431_nth__rule,axiom,
    ! [I: nat,Xs2: list_nat,A: array_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_nth_nat @ A @ I )
        @ ^ [R2: nat] :
            ( times_times_assn @ ( snga_assn_nat @ A @ Xs2 )
            @ ( pure_assn
              @ ( R2
                = ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_4432_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( hoare_3904069481286416050_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_8141364883501958055_VEBTi @ A )
      @ ^ [R2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( pure_assn @ ( R2 = Xs2 ) ) ) ) ).

% freeze_rule
thf(fact_4433_upd__rule,axiom,
    ! [I: nat,Xs2: list_o,A: array_o,X4: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( hoare_6478655245392655262rray_o @ ( snga_assn_o @ A @ Xs2 ) @ ( array_upd_o @ I @ X4 @ A )
        @ ^ [R2: array_o] : ( times_times_assn @ ( snga_assn_o @ A @ ( list_update_o @ Xs2 @ I @ X4 ) ) @ ( pure_assn @ ( R2 = A ) ) ) ) ) ).

% upd_rule
thf(fact_4434_upd__rule,axiom,
    ! [I: nat,Xs2: list_nat,A: array_nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( hoare_6807272225193264096ay_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_upd_nat @ I @ X4 @ A )
        @ ^ [R2: array_nat] : ( times_times_assn @ ( snga_assn_nat @ A @ ( list_update_nat @ Xs2 @ I @ X4 ) ) @ ( pure_assn @ ( R2 = A ) ) ) ) ) ).

% upd_rule
thf(fact_4435_upd__rule,axiom,
    ! [I: nat,Xs2: list_int,A: array_int,X4: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( hoare_2629421205684067388ay_int @ ( snga_assn_int @ A @ Xs2 ) @ ( array_upd_int @ I @ X4 @ A )
        @ ^ [R2: array_int] : ( times_times_assn @ ( snga_assn_int @ A @ ( list_update_int @ Xs2 @ I @ X4 ) ) @ ( pure_assn @ ( R2 = A ) ) ) ) ) ).

% upd_rule
thf(fact_4436_upd__rule,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A: array_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( hoare_3353465787467722821_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_upd_VEBT_VEBTi @ I @ X4 @ A )
        @ ^ [R2: array_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X4 ) ) @ ( pure_assn @ ( R2 = A ) ) ) ) ) ).

% upd_rule
thf(fact_4437_listI__assn__def,axiom,
    ( vEBT_L886525131989349516_VEBTi
    = ( ^ [I7: set_nat,A5: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,Xsi3: list_VEBT_VEBTi] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s7982070591426661849_VEBTi @ Xsi3 )
                = ( size_s7982070591426661849_VEBTi @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4438_listI__assn__def,axiom,
    ( vEBT_L2497118539674116125T_VEBT
    = ( ^ [I7: set_nat,A5: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,Xsi3: list_VEBT_VEBT] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s6755466524823107622T_VEBT @ Xsi3 )
                = ( size_s7982070591426661849_VEBTi @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4439_listI__assn__def,axiom,
    ( vEBT_L3204528365124325536T_VEBT
    = ( ^ [I7: set_nat,A5: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,Xsi3: list_VEBT_VEBT] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s6755466524823107622T_VEBT @ Xsi3 )
                = ( size_s6755466524823107622T_VEBT @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4440_listI__assn__def,axiom,
    ( vEBT_L7851252805511451907_VEBTi
    = ( ^ [I7: set_nat,A5: real > vEBT_VEBTi > assn,Xs: list_real,Xsi3: list_VEBT_VEBTi] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s7982070591426661849_VEBTi @ Xsi3 )
                = ( size_size_list_real @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_real @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4441_listI__assn__def,axiom,
    ( vEBT_L3095048238742455910T_VEBT
    = ( ^ [I7: set_nat,A5: real > vEBT_VEBT > assn,Xs: list_real,Xsi3: list_VEBT_VEBT] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s6755466524823107622T_VEBT @ Xsi3 )
                = ( size_size_list_real @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_real @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4442_listI__assn__def,axiom,
    ( vEBT_L6286945158656146733_VEBTi
    = ( ^ [I7: set_nat,A5: $o > vEBT_VEBTi > assn,Xs: list_o,Xsi3: list_VEBT_VEBTi] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s7982070591426661849_VEBTi @ Xsi3 )
                = ( size_size_list_o @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_o @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4443_listI__assn__def,axiom,
    ( vEBT_L1319876754960170684T_VEBT
    = ( ^ [I7: set_nat,A5: $o > vEBT_VEBT > assn,Xs: list_o,Xsi3: list_VEBT_VEBT] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s6755466524823107622T_VEBT @ Xsi3 )
                = ( size_size_list_o @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_o @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4444_listI__assn__def,axiom,
    ( vEBT_L7489483478785760935_VEBTi
    = ( ^ [I7: set_nat,A5: nat > vEBT_VEBTi > assn,Xs: list_nat,Xsi3: list_VEBT_VEBTi] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s7982070591426661849_VEBTi @ Xsi3 )
                = ( size_size_list_nat @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_nat @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4445_listI__assn__def,axiom,
    ( vEBT_L8511957252848910786T_VEBT
    = ( ^ [I7: set_nat,A5: nat > vEBT_VEBT > assn,Xs: list_nat,Xsi3: list_VEBT_VEBT] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s6755466524823107622T_VEBT @ Xsi3 )
                = ( size_size_list_nat @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_nat @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4446_listI__assn__def,axiom,
    ( vEBT_L114188773329725699_VEBTi
    = ( ^ [I7: set_nat,A5: int > vEBT_VEBTi > assn,Xs: list_int,Xsi3: list_VEBT_VEBTi] :
          ( times_times_assn
          @ ( pure_assn
            @ ( ( ( size_s7982070591426661849_VEBTi @ Xsi3 )
                = ( size_size_list_int @ Xs ) )
              & ( ord_less_eq_set_nat @ I7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ) )
          @ ( finite_fold_nat_assn
            @ ^ [I3: nat,A2: assn] : ( times_times_assn @ A2 @ ( A5 @ ( nth_int @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi3 @ I3 ) ) )
            @ one_one_assn
            @ I7 ) ) ) ) ).

% listI_assn_def
thf(fact_4447_length__rule,axiom,
    ! [A: array_o,Xs2: list_o] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_o @ A @ Xs2 ) @ ( array_len_o @ A )
      @ ^ [R2: nat] :
          ( times_times_assn @ ( snga_assn_o @ A @ Xs2 )
          @ ( pure_assn
            @ ( R2
              = ( size_size_list_o @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_4448_length__rule,axiom,
    ! [A: array_nat,Xs2: list_nat] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs2 ) @ ( array_len_nat @ A )
      @ ^ [R2: nat] :
          ( times_times_assn @ ( snga_assn_nat @ A @ Xs2 )
          @ ( pure_assn
            @ ( R2
              = ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_4449_length__rule,axiom,
    ! [A: array_int,Xs2: list_int] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_int @ A @ Xs2 ) @ ( array_len_int @ A )
      @ ^ [R2: nat] :
          ( times_times_assn @ ( snga_assn_int @ A @ Xs2 )
          @ ( pure_assn
            @ ( R2
              = ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_4450_length__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 ) @ ( array_len_VEBT_VEBTi @ A )
      @ ^ [R2: nat] :
          ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs2 )
          @ ( pure_assn
            @ ( R2
              = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ) ) ).

% length_rule
thf(fact_4451_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > nat,X4: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( distinct_VEBT_VEBT @ L )
       => ( ( list_update_nat @ ( map_VEBT_VEBT_nat @ F @ L ) @ I @ X4 )
          = ( map_VEBT_VEBT_nat @ ( fun_up6512855943550542919BT_nat @ F @ ( nth_VEBT_VEBT @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4452_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > real,X4: real] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( distinct_VEBT_VEBT @ L )
       => ( ( list_update_real @ ( map_VEBT_VEBT_real @ F @ L ) @ I @ X4 )
          = ( map_VEBT_VEBT_real @ ( fun_up7749720967766031267T_real @ F @ ( nth_VEBT_VEBT @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4453_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( distinct_VEBT_VEBTi @ L )
       => ( ( list_u6098035379799741383_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ L ) @ I @ X4 )
          = ( map_VE483055756984248624_VEBTi @ ( fun_up8301472745780153321_VEBTi @ F @ ( nth_VEBT_VEBTi @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4454_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( distinct_VEBT_VEBT @ L )
       => ( ( list_u6098035379799741383_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ L ) @ I @ X4 )
          = ( map_VE7029150624388687525_VEBTi @ ( fun_up377479910058727724_VEBTi @ F @ ( nth_VEBT_VEBT @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4455_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( distinct_VEBT_VEBTi @ L )
       => ( ( list_u1324408373059187874T_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ L ) @ I @ X4 )
          = ( map_VE7998069337340375161T_VEBT @ ( fun_up1346398623010415360T_VEBT @ F @ ( nth_VEBT_VEBTi @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4456_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( distinct_VEBT_VEBT @ L )
       => ( ( list_u1324408373059187874T_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ L ) @ I @ X4 )
          = ( map_VE8901447254227204932T_VEBT @ ( fun_up224749957652071293T_VEBT @ F @ ( nth_VEBT_VEBT @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4457_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_real,F: real > vEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( distinct_real @ L )
       => ( ( list_u6098035379799741383_VEBTi @ ( map_real_VEBT_VEBTi @ F @ L ) @ I @ X4 )
          = ( map_real_VEBT_VEBTi @ ( fun_up8049757234741329632_VEBTi @ F @ ( nth_real @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4458_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_real,F: real > vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( distinct_real @ L )
       => ( ( list_u1324408373059187874T_VEBT @ ( map_real_VEBT_VEBT @ F @ L ) @ I @ X4 )
          = ( map_real_VEBT_VEBT @ ( fun_up6563732700392937161T_VEBT @ F @ ( nth_real @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4459_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_o,F: $o > vEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( distinct_o @ L )
       => ( ( list_u6098035379799741383_VEBTi @ ( map_o_VEBT_VEBTi @ F @ L ) @ I @ X4 )
          = ( map_o_VEBT_VEBTi @ ( fun_upd_o_VEBT_VEBTi @ F @ ( nth_o @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4460_map__distinct__upd__conv,axiom,
    ! [I: nat,L: list_o,F: $o > vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( distinct_o @ L )
       => ( ( list_u1324408373059187874T_VEBT @ ( map_o_VEBT_VEBT @ F @ L ) @ I @ X4 )
          = ( map_o_VEBT_VEBT @ ( fun_upd_o_VEBT_VEBT @ F @ ( nth_o @ L @ I ) @ X4 ) @ L ) ) ) ) ).

% map_distinct_upd_conv
thf(fact_4461_horner__sum__foldr,axiom,
    ( groups1503878375050959669l_real
    = ( ^ [F4: real > real,A2: real,Xs: list_real] :
          ( foldr_real_real
          @ ^ [X: real,B2: real] : ( plus_plus_real @ ( F4 @ X ) @ ( times_times_real @ A2 @ B2 ) )
          @ Xs
          @ zero_zero_real ) ) ) ).

% horner_sum_foldr
thf(fact_4462_horner__sum__foldr,axiom,
    ( groups7488368174851004413at_nat
    = ( ^ [F4: nat > nat,A2: nat,Xs: list_nat] :
          ( foldr_nat_nat
          @ ^ [X: nat,B2: nat] : ( plus_plus_nat @ ( F4 @ X ) @ ( times_times_nat @ A2 @ B2 ) )
          @ Xs
          @ zero_zero_nat ) ) ) ).

% horner_sum_foldr
thf(fact_4463_horner__sum__foldr,axiom,
    ( groups9116527308978886569_o_int
    = ( ^ [F4: $o > int,A2: int,Xs: list_o] :
          ( foldr_o_int
          @ ^ [X: $o,B2: int] : ( plus_plus_int @ ( F4 @ X ) @ ( times_times_int @ A2 @ B2 ) )
          @ Xs
          @ zero_zero_int ) ) ) ).

% horner_sum_foldr
thf(fact_4464_set__remove1__eq,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) )
        = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_remove1_eq
thf(fact_4465_set__remove1__eq,axiom,
    ! [Xs2: list_real,X4: real] :
      ( ( distinct_real @ Xs2 )
     => ( ( set_real2 @ ( remove1_real @ X4 @ Xs2 ) )
        = ( minus_minus_set_real @ ( set_real2 @ Xs2 ) @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% set_remove1_eq
thf(fact_4466_set__remove1__eq,axiom,
    ! [Xs2: list_o,X4: $o] :
      ( ( distinct_o @ Xs2 )
     => ( ( set_o2 @ ( remove1_o @ X4 @ Xs2 ) )
        = ( minus_minus_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% set_remove1_eq
thf(fact_4467_set__remove1__eq,axiom,
    ! [Xs2: list_int,X4: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( set_int2 @ ( remove1_int @ X4 @ Xs2 ) )
        = ( minus_minus_set_int @ ( set_int2 @ Xs2 ) @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% set_remove1_eq
thf(fact_4468_set__remove1__eq,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( set_nat2 @ ( remove1_nat @ X4 @ Xs2 ) )
        = ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_4469_listsum__bound,axiom,
    ! [Xs2: list_int,F: int > real,Y: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4470_listsum__bound,axiom,
    ! [Xs2: list_set_nat,F: set_nat > real,Y: real] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_set_nat_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4471_listsum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Y: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4472_listsum__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,Y: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4473_listsum__bound,axiom,
    ! [Xs2: list_real,F: real > real,Y: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4474_listsum__bound,axiom,
    ! [Xs2: list_o,F: $o > real,Y: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4475_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,C: real,G: vEBT_VEBT > real,D: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4476_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,C: real,G: nat > real,D: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4477_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_real,F: real > real,C: real,G: real > real,D: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4478_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_o,F: $o > real,C: real,G: $o > real,D: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4479_set__remove1__subset,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% set_remove1_subset
thf(fact_4480_set__remove1__subset,axiom,
    ! [X4: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X4 @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).

% set_remove1_subset
thf(fact_4481_set__remove1__subset,axiom,
    ! [X4: real,Xs2: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( remove1_real @ X4 @ Xs2 ) ) @ ( set_real2 @ Xs2 ) ) ).

% set_remove1_subset
thf(fact_4482_set__remove1__subset,axiom,
    ! [X4: $o,Xs2: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( remove1_o @ X4 @ Xs2 ) ) @ ( set_o2 @ Xs2 ) ) ).

% set_remove1_subset
thf(fact_4483_set__remove1__subset,axiom,
    ! [X4: int,Xs2: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( remove1_int @ X4 @ Xs2 ) ) @ ( set_int2 @ Xs2 ) ) ).

% set_remove1_subset
thf(fact_4484_sorted__remove1,axiom,
    ! [Xs2: list_rat,A: rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ ( remove1_rat @ A @ Xs2 ) ) ) ).

% sorted_remove1
thf(fact_4485_sorted__remove1,axiom,
    ! [Xs2: list_num,A: num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( sorted_wrt_num @ ord_less_eq_num @ ( remove1_num @ A @ Xs2 ) ) ) ).

% sorted_remove1
thf(fact_4486_sorted__remove1,axiom,
    ! [Xs2: list_nat,A: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs2 ) ) ) ).

% sorted_remove1
thf(fact_4487_sorted__remove1,axiom,
    ! [Xs2: list_int,A: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( remove1_int @ A @ Xs2 ) ) ) ).

% sorted_remove1
thf(fact_4488_sorted__map__remove1,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
     => ( sorted_wrt_real @ ord_less_eq_real @ ( map_VEBT_VEBT_real @ F @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) ) ) ) ).

% sorted_map_remove1
thf(fact_4489_sorted__map__remove1,axiom,
    ! [F: product_prod_o_o > $o,Xs2: list_P4002435161011370285od_o_o,X4: product_prod_o_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( map_Pr7541730621154948341_o_o_o @ F @ Xs2 ) )
     => ( sorted_wrt_o @ ord_less_eq_o @ ( map_Pr7541730621154948341_o_o_o @ F @ ( remove8878500798450800835od_o_o @ X4 @ Xs2 ) ) ) ) ).

% sorted_map_remove1
thf(fact_4490_sorted__map__remove1,axiom,
    ! [F: nat > $o,Xs2: list_nat,X4: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( map_nat_o @ F @ Xs2 ) )
     => ( sorted_wrt_o @ ord_less_eq_o @ ( map_nat_o @ F @ ( remove1_nat @ X4 @ Xs2 ) ) ) ) ).

% sorted_map_remove1
thf(fact_4491_sorted__map__remove1,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_VEBT_VEBT_nat @ F @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) ) ) ) ).

% sorted_map_remove1
thf(fact_4492_sorted__map__remove1,axiom,
    ! [F: nat > nat,Xs2: list_nat,X4: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( remove1_nat @ X4 @ Xs2 ) ) ) ) ).

% sorted_map_remove1
thf(fact_4493_length__remove1,axiom,
    ! [X4: set_nat,Xs2: list_set_nat] :
      ( ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
       => ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
       => ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X4 @ Xs2 ) )
          = ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4494_length__remove1,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( ( size_s6755466524823107622T_VEBT @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( ( size_s6755466524823107622T_VEBT @ ( remove1_VEBT_VEBT @ X4 @ Xs2 ) )
          = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4495_length__remove1,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ( ( size_size_list_real @ ( remove1_real @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_size_list_real @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ( ( size_size_list_real @ ( remove1_real @ X4 @ Xs2 ) )
          = ( size_size_list_real @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4496_length__remove1,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ( ( size_size_list_o @ ( remove1_o @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ( ( size_size_list_o @ ( remove1_o @ X4 @ Xs2 ) )
          = ( size_size_list_o @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4497_length__remove1,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X4 @ Xs2 ) )
          = ( size_size_list_nat @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4498_length__remove1,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
       => ( ( size_size_list_int @ ( remove1_int @ X4 @ Xs2 ) )
          = ( minus_minus_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
       => ( ( size_size_list_int @ ( remove1_int @ X4 @ Xs2 ) )
          = ( size_size_list_int @ Xs2 ) ) ) ) ).

% length_remove1
thf(fact_4499_real__nat__list,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_VEBT_VEBT_real
          @ ^ [X: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( F @ X ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_4500_real__nat__list,axiom,
    ! [F: nat > nat,Xs2: list_nat,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_nat_real
          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_4501_new__rule,axiom,
    ! [N: nat,X4: $o] :
      ( hoare_6478655245392655262rray_o @ one_one_assn @ ( array_new_o @ N @ X4 )
      @ ^ [R2: array_o] : ( snga_assn_o @ R2 @ ( replicate_o @ N @ X4 ) ) ) ).

% new_rule
thf(fact_4502_new__rule,axiom,
    ! [N: nat,X4: vEBT_VEBTi] :
      ( hoare_3353465787467722821_VEBTi @ one_one_assn @ ( array_new_VEBT_VEBTi @ N @ X4 )
      @ ^ [R2: array_VEBT_VEBTi] : ( snga_assn_VEBT_VEBTi @ R2 @ ( replicate_VEBT_VEBTi @ N @ X4 ) ) ) ).

% new_rule
thf(fact_4503_aux,axiom,
    ! [P: vEBT_VEBTi > vEBT_VEBTi > assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s7982070591426661849_VEBTi @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ As @ I3 ) @ ( nth_VEBT_VEBTi @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ As ) ) ) ) ) ).

% aux
thf(fact_4504_aux,axiom,
    ! [P: vEBT_VEBTi > vEBT_VEBT > assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: vEBT_VEBT,Cs: list_VEBT_VEBT] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s7982070591426661849_VEBTi @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ As @ I3 ) @ ( nth_VEBT_VEBT @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ As ) ) ) ) ) ).

% aux
thf(fact_4505_aux,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s6755466524823107622T_VEBT @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ As @ I3 ) @ ( nth_VEBT_VEBTi @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ As ) ) ) ) ) ).

% aux
thf(fact_4506_aux,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBT > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBT,Cs: list_VEBT_VEBT] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s6755466524823107622T_VEBT @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ As @ I3 ) @ ( nth_VEBT_VEBT @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ As ) ) ) ) ) ).

% aux
thf(fact_4507_aux,axiom,
    ! [P: vEBT_VEBTi > nat > assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: nat,Cs: list_nat] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ A @ As ) @ I3 ) @ ( nth_nat @ ( cons_nat @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s7982070591426661849_VEBTi @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ As @ I3 ) @ ( nth_nat @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ As ) ) ) ) ) ).

% aux
thf(fact_4508_aux,axiom,
    ! [P: vEBT_VEBT > nat > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: nat,Cs: list_nat] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ A @ As ) @ I3 ) @ ( nth_nat @ ( cons_nat @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s6755466524823107622T_VEBT @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ As @ I3 ) @ ( nth_nat @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ As ) ) ) ) ) ).

% aux
thf(fact_4509_aux,axiom,
    ! [P: vEBT_VEBTi > int > assn,A: vEBT_VEBTi,As: list_VEBT_VEBTi,C: int,Cs: list_int] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ A @ As ) @ I3 ) @ ( nth_int @ ( cons_int @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s7982070591426661849_VEBTi @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBTi @ As @ I3 ) @ ( nth_int @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ As ) ) ) ) ) ).

% aux
thf(fact_4510_aux,axiom,
    ! [P: vEBT_VEBT > int > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: int,Cs: list_int] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ A @ As ) @ I3 ) @ ( nth_int @ ( cons_int @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_s6755466524823107622T_VEBT @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_VEBT_VEBT @ As @ I3 ) @ ( nth_int @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ As ) ) ) ) ) ).

% aux
thf(fact_4511_aux,axiom,
    ! [P: real > vEBT_VEBTi > assn,A: real,As: list_real,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_real @ ( cons_real @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_size_list_real @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_real @ As @ I3 ) @ ( nth_VEBT_VEBTi @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ As ) ) ) ) ) ).

% aux
thf(fact_4512_aux,axiom,
    ! [P: real > vEBT_VEBT > assn,A: real,As: list_real,C: vEBT_VEBT,Cs: list_VEBT_VEBT] :
      ( ( finite_fold_nat_assn
        @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_real @ ( cons_real @ A @ As ) @ I3 ) @ ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ C @ Cs ) @ I3 ) ) )
        @ one_one_assn
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ ( size_size_list_real @ As ) ) ) )
      = ( times_times_assn @ ( P @ A @ C )
        @ ( finite_fold_nat_assn
          @ ^ [I3: nat,Aa: assn] : ( times_times_assn @ Aa @ ( P @ ( nth_real @ As @ I3 ) @ ( nth_VEBT_VEBT @ Cs @ I3 ) ) )
          @ one_one_assn
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ As ) ) ) ) ) ).

% aux
thf(fact_4513_of__list__rule,axiom,
    ! [Xs2: list_VEBT_VEBTi] :
      ( hoare_3353465787467722821_VEBTi @ one_one_assn @ ( array_615059503499738533_VEBTi @ Xs2 )
      @ ^ [R2: array_VEBT_VEBTi] : ( snga_assn_VEBT_VEBTi @ R2 @ Xs2 ) ) ).

% of_list_rule
thf(fact_4514_foldr0,axiom,
    ! [Xs2: list_real,C: real,D: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs2 @ ( plus_plus_real @ C @ D ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs2 @ D ) @ C ) ) ).

% foldr0
thf(fact_4515_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_4516_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_4517_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_4518_foldr__same,axiom,
    ! [Xs2: list_real,Y: real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs2 @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Y ) ) ) ) ).

% foldr_same
thf(fact_4519_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4520_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_real,F: real > real,Bound: real,I: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4521_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_o,F: $o > real,Bound: real,I: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_o @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4522_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_nat,F: nat > real,Bound: real,I: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4523_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_int,F: int > real,Bound: real,I: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_int @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4524_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_4525_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_4526_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_4527_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_4528_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_4529_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_4530_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_4531_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_4532_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_4533_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_4534_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% semiring_1_class.of_nat_0
thf(fact_4535_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_4536_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_4537_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_4538_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_4539_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_4540_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_4541_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_4542_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_4543_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_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_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_4551_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_4552_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_4553_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_4554_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_4555_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_4556_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_4557_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_4558_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_4559_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_4560_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_4561_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_4562_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_4563_nth__Cons__Suc,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,N: nat] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_4564_nth__Cons__Suc,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBT @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_4565_nth__Cons__Suc,axiom,
    ! [X4: nat,Xs2: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_nat @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_4566_nth__Cons__Suc,axiom,
    ! [X4: int,Xs2: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_int @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_4567_length__nth__simps_I4_J,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,N: nat] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBTi @ Xs2 @ N ) ) ).

% length_nth_simps(4)
thf(fact_4568_length__nth__simps_I4_J,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBT @ Xs2 @ N ) ) ).

% length_nth_simps(4)
thf(fact_4569_length__nth__simps_I4_J,axiom,
    ! [X4: nat,Xs2: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_nat @ Xs2 @ N ) ) ).

% length_nth_simps(4)
thf(fact_4570_length__nth__simps_I4_J,axiom,
    ! [X4: int,Xs2: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ ( suc @ N ) )
      = ( nth_int @ Xs2 @ N ) ) ).

% length_nth_simps(4)
thf(fact_4571_nth__Cons__0,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% nth_Cons_0
thf(fact_4572_nth__Cons__0,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% nth_Cons_0
thf(fact_4573_nth__Cons__0,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% nth_Cons_0
thf(fact_4574_nth__Cons__0,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% nth_Cons_0
thf(fact_4575_length__nth__simps_I3_J,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% length_nth_simps(3)
thf(fact_4576_length__nth__simps_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% length_nth_simps(3)
thf(fact_4577_length__nth__simps_I3_J,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% length_nth_simps(3)
thf(fact_4578_length__nth__simps_I3_J,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ zero_zero_nat )
      = X4 ) ).

% length_nth_simps(3)
thf(fact_4579_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_4580_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_4581_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_4582_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_4583_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_4584_foldl__length,axiom,
    ! [L: list_real] :
      ( ( foldl_nat_real
        @ ^ [I3: nat,X: real] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_real @ L ) ) ).

% foldl_length
thf(fact_4585_foldl__length,axiom,
    ! [L: list_o] :
      ( ( foldl_nat_o
        @ ^ [I3: nat,X: $o] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_o @ L ) ) ).

% foldl_length
thf(fact_4586_foldl__length,axiom,
    ! [L: list_nat] :
      ( ( foldl_nat_nat
        @ ^ [I3: nat,X: nat] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_nat @ L ) ) ).

% foldl_length
thf(fact_4587_foldl__length,axiom,
    ! [L: list_int] :
      ( ( foldl_nat_int
        @ ^ [I3: nat,X: int] : ( suc @ I3 )
        @ zero_zero_nat
        @ L )
      = ( size_size_list_int @ L ) ) ).

% foldl_length
thf(fact_4588_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_4589_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_4590_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_4591_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_4592_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_4593_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_4594_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_4595_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_4596_horner__sum__simps_I2_J,axiom,
    ! [F: nat > real,A: real,X4: nat,Xs2: list_nat] :
      ( ( groups3482786445295563865t_real @ F @ A @ ( cons_nat @ X4 @ Xs2 ) )
      = ( plus_plus_real @ ( F @ X4 ) @ ( times_times_real @ A @ ( groups3482786445295563865t_real @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4597_horner__sum__simps_I2_J,axiom,
    ! [F: int > real,A: real,X4: int,Xs2: list_int] :
      ( ( groups5669708019988585653t_real @ F @ A @ ( cons_int @ X4 @ Xs2 ) )
      = ( plus_plus_real @ ( F @ X4 ) @ ( times_times_real @ A @ ( groups5669708019988585653t_real @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4598_horner__sum__simps_I2_J,axiom,
    ! [F: nat > rat,A: rat,X4: nat,Xs2: list_nat] :
      ( ( groups6853238114764508677at_rat @ F @ A @ ( cons_nat @ X4 @ Xs2 ) )
      = ( plus_plus_rat @ ( F @ X4 ) @ ( times_times_rat @ A @ ( groups6853238114764508677at_rat @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4599_horner__sum__simps_I2_J,axiom,
    ! [F: int > rat,A: rat,X4: int,Xs2: list_int] :
      ( ( groups7852591826665563233nt_rat @ F @ A @ ( cons_int @ X4 @ Xs2 ) )
      = ( plus_plus_rat @ ( F @ X4 ) @ ( times_times_rat @ A @ ( groups7852591826665563233nt_rat @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4600_horner__sum__simps_I2_J,axiom,
    ! [F: nat > nat,A: nat,X4: nat,Xs2: list_nat] :
      ( ( groups7488368174851004413at_nat @ F @ A @ ( cons_nat @ X4 @ Xs2 ) )
      = ( plus_plus_nat @ ( F @ X4 ) @ ( times_times_nat @ A @ ( groups7488368174851004413at_nat @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4601_horner__sum__simps_I2_J,axiom,
    ! [F: int > nat,A: nat,X4: int,Xs2: list_int] :
      ( ( groups8487721886752058969nt_nat @ F @ A @ ( cons_int @ X4 @ Xs2 ) )
      = ( plus_plus_nat @ ( F @ X4 ) @ ( times_times_nat @ A @ ( groups8487721886752058969nt_nat @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4602_horner__sum__simps_I2_J,axiom,
    ! [F: nat > int,A: int,X4: nat,Xs2: list_nat] :
      ( ( groups7485877704341954137at_int @ F @ A @ ( cons_nat @ X4 @ Xs2 ) )
      = ( plus_plus_int @ ( F @ X4 ) @ ( times_times_int @ A @ ( groups7485877704341954137at_int @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4603_horner__sum__simps_I2_J,axiom,
    ! [F: int > int,A: int,X4: int,Xs2: list_int] :
      ( ( groups8485231416243008693nt_int @ F @ A @ ( cons_int @ X4 @ Xs2 ) )
      = ( plus_plus_int @ ( F @ X4 ) @ ( times_times_int @ A @ ( groups8485231416243008693nt_int @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4604_horner__sum__simps_I2_J,axiom,
    ! [F: nat > complex,A: complex,X4: nat,Xs2: list_nat] :
      ( ( groups404637655443745499omplex @ F @ A @ ( cons_nat @ X4 @ Xs2 ) )
      = ( plus_plus_complex @ ( F @ X4 ) @ ( times_times_complex @ A @ ( groups404637655443745499omplex @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4605_horner__sum__simps_I2_J,axiom,
    ! [F: int > complex,A: complex,X4: int,Xs2: list_int] :
      ( ( groups1380173120649922871omplex @ F @ A @ ( cons_int @ X4 @ Xs2 ) )
      = ( plus_plus_complex @ ( F @ X4 ) @ ( times_times_complex @ A @ ( groups1380173120649922871omplex @ F @ A @ Xs2 ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_4606_enumerate__simps_I2_J,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( enumerate_int @ N @ ( cons_int @ X4 @ Xs2 ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X4 ) @ ( enumerate_int @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_4607_enumerate__simps_I2_J,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X4 @ Xs2 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X4 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_4608_enumerate__simps_I2_J,axiom,
    ! [N: nat,X4: num,Xs2: list_num] :
      ( ( enumerate_num @ N @ ( cons_num @ X4 @ Xs2 ) )
      = ( cons_P7221140103528650497at_num @ ( product_Pair_nat_num @ N @ X4 ) @ ( enumerate_num @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_4609_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_4610_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_4611_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_4612_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_4613_nth__Cons__pos,axiom,
    ! [N: nat,X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ N )
        = ( nth_VEBT_VEBTi @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4614_nth__Cons__pos,axiom,
    ! [N: nat,X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ N )
        = ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4615_nth__Cons__pos,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ N )
        = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4616_nth__Cons__pos,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ N )
        = ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4617_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X6: real] : ( member_real @ X6 @ S3 )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ S3 )
               => ( ord_less_eq_real @ X6 @ 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_4618_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y4: real] :
          ( ( ord_less_real @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% less_eq_real_def
thf(fact_4619_reals__Archimedean3,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ! [Y5: real] :
        ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) ) ) ).

% reals_Archimedean3
thf(fact_4620_product__lists_Osimps_I2_J,axiom,
    ! [Xs2: list_nat,Xss: list_list_nat] :
      ( ( product_lists_nat @ ( cons_list_nat @ Xs2 @ Xss ) )
      = ( concat_list_nat
        @ ( map_na6205611841492582150st_nat
          @ ^ [X: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( product_lists_nat @ Xss ) )
          @ Xs2 ) ) ) ).

% product_lists.simps(2)
thf(fact_4621_product__lists_Osimps_I2_J,axiom,
    ! [Xs2: list_int,Xss: list_list_int] :
      ( ( product_lists_int @ ( cons_list_int @ Xs2 @ Xss ) )
      = ( concat_list_int
        @ ( map_in1039701548059952062st_int
          @ ^ [X: int] : ( map_li4896172289311737022st_int @ ( cons_int @ X ) @ ( product_lists_int @ Xss ) )
          @ Xs2 ) ) ) ).

% product_lists.simps(2)
thf(fact_4622_real__arch__simple,axiom,
    ! [X4: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_4623_real__arch__simple,axiom,
    ! [X4: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X4 @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_4624_reals__Archimedean2,axiom,
    ! [X4: rat] :
    ? [N2: nat] : ( ord_less_rat @ X4 @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_4625_reals__Archimedean2,axiom,
    ! [X4: real] :
    ? [N2: nat] : ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_4626_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X4 ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4627_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X4 ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4628_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4629_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4630_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4631_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_4632_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_4633_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_4634_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_4635_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_4636_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_4637_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_4638_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_4639_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_4640_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_4641_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_4642_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_4643_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_4644_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_4645_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_4646_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_4647_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_4648_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_4649_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_4650_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_4651_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_4652_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_4653_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_4654_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_4655_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_4656_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_4657_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_4658_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_rat @ ( semiri681578069525770553at_rat @ A ) @ ( semiri681578069525770553at_rat @ B ) )
      = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4659_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4660_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4661_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ A ) @ ( semiri1316708129612266289at_nat @ B ) )
      = ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4662_real__archimedian__rdiv__eq__0,axiom,
    ! [X4: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M2 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X4 ) @ C ) )
         => ( X4 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_4663_length__nth__simps_I2_J,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( size_size_list_real @ ( cons_real @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_4664_length__nth__simps_I2_J,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_4665_length__nth__simps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_4666_length__nth__simps_I2_J,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_4667_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ( suc @ N )
        = ( size_size_list_real @ Xs2 ) )
      = ( ? [Y4: real,Ys3: list_real] :
            ( ( Xs2
              = ( cons_real @ Y4 @ Ys3 ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4668_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ( suc @ N )
        = ( size_size_list_o @ Xs2 ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4669_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs2 ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4670_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs2 ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4671_length__Suc__conv,axiom,
    ! [Xs2: list_real,N: nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: real,Ys3: list_real] :
            ( ( Xs2
              = ( cons_real @ Y4 @ Ys3 ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4672_length__Suc__conv,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4673_length__Suc__conv,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4674_length__Suc__conv,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4675_impossible__Cons,axiom,
    ! [Xs2: list_real,Ys: list_real,X4: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) )
     => ( Xs2
       != ( cons_real @ X4 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4676_impossible__Cons,axiom,
    ! [Xs2: list_o,Ys: list_o,X4: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) )
     => ( Xs2
       != ( cons_o @ X4 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4677_impossible__Cons,axiom,
    ! [Xs2: list_nat,Ys: list_nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs2
       != ( cons_nat @ X4 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4678_impossible__Cons,axiom,
    ! [Xs2: list_int,Ys: list_int,X4: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) )
     => ( Xs2
       != ( cons_int @ X4 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4679_set__subset__Cons,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_4680_set__subset__Cons,axiom,
    ! [Xs2: list_real,X4: real] : ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ ( set_real2 @ ( cons_real @ X4 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_4681_set__subset__Cons,axiom,
    ! [Xs2: list_o,X4: $o] : ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ ( set_o2 @ ( cons_o @ X4 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_4682_set__subset__Cons,axiom,
    ! [Xs2: list_nat,X4: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X4 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_4683_set__subset__Cons,axiom,
    ! [Xs2: list_int,X4: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ ( set_int2 @ ( cons_int @ X4 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_4684_sorted2,axiom,
    ! [X4: rat,Y: rat,Zs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ X4 @ ( cons_rat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_rat @ X4 @ Y )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4685_sorted2,axiom,
    ! [X4: num,Y: num,Zs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ X4 @ ( cons_num @ Y @ Zs ) ) )
      = ( ( ord_less_eq_num @ X4 @ Y )
        & ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4686_sorted2,axiom,
    ! [X4: nat,Y: nat,Zs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X4 @ ( cons_nat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_nat @ X4 @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4687_sorted2,axiom,
    ! [X4: int,Y: int,Zs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X4 @ ( cons_int @ Y @ Zs ) ) )
      = ( ( ord_less_eq_int @ X4 @ Y )
        & ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4688_list__update__code_I3_J,axiom,
    ! [X4: nat,Xs2: list_nat,I: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( suc @ I ) @ Y )
      = ( cons_nat @ X4 @ ( list_update_nat @ Xs2 @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4689_list__update__code_I3_J,axiom,
    ! [X4: int,Xs2: list_int,I: nat,Y: int] :
      ( ( list_update_int @ ( cons_int @ X4 @ Xs2 ) @ ( suc @ I ) @ Y )
      = ( cons_int @ X4 @ ( list_update_int @ Xs2 @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4690_list__update__code_I3_J,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ ( suc @ I ) @ Y )
      = ( cons_VEBT_VEBTi @ X4 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4691_list__update__code_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ ( suc @ I ) @ Y )
      = ( cons_VEBT_VEBT @ X4 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4692_list__update__code_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X4 @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4693_list__update__code_I2_J,axiom,
    ! [X4: int,Xs2: list_int,Y: int] :
      ( ( list_update_int @ ( cons_int @ X4 @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_int @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4694_list__update__code_I2_J,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBTi @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4695_list__update__code_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBT @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4696_replicate__Suc,axiom,
    ! [N: nat,X4: nat] :
      ( ( replicate_nat @ ( suc @ N ) @ X4 )
      = ( cons_nat @ X4 @ ( replicate_nat @ N @ X4 ) ) ) ).

% replicate_Suc
thf(fact_4697_replicate__Suc,axiom,
    ! [N: nat,X4: int] :
      ( ( replicate_int @ ( suc @ N ) @ X4 )
      = ( cons_int @ X4 @ ( replicate_int @ N @ X4 ) ) ) ).

% replicate_Suc
thf(fact_4698_replicate__Suc,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT @ ( suc @ N ) @ X4 )
      = ( cons_VEBT_VEBT @ X4 @ ( replicate_VEBT_VEBT @ N @ X4 ) ) ) ).

% replicate_Suc
thf(fact_4699_replicate__Suc,axiom,
    ! [N: nat,X4: $o] :
      ( ( replicate_o @ ( suc @ N ) @ X4 )
      = ( cons_o @ X4 @ ( replicate_o @ N @ X4 ) ) ) ).

% replicate_Suc
thf(fact_4700_foldl__absorb1,axiom,
    ! [X4: real,Zs: list_real] :
      ( ( times_times_real @ X4 @ ( foldl_real_real @ times_times_real @ one_one_real @ Zs ) )
      = ( foldl_real_real @ times_times_real @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4701_foldl__absorb1,axiom,
    ! [X4: rat,Zs: list_rat] :
      ( ( times_times_rat @ X4 @ ( foldl_rat_rat @ times_times_rat @ one_one_rat @ Zs ) )
      = ( foldl_rat_rat @ times_times_rat @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4702_foldl__absorb1,axiom,
    ! [X4: nat,Zs: list_nat] :
      ( ( times_times_nat @ X4 @ ( foldl_nat_nat @ times_times_nat @ one_one_nat @ Zs ) )
      = ( foldl_nat_nat @ times_times_nat @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4703_foldl__absorb1,axiom,
    ! [X4: int,Zs: list_int] :
      ( ( times_times_int @ X4 @ ( foldl_int_int @ times_times_int @ one_one_int @ Zs ) )
      = ( foldl_int_int @ times_times_int @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4704_foldl__absorb1,axiom,
    ! [X4: assn,Zs: list_assn] :
      ( ( times_times_assn @ X4 @ ( foldl_assn_assn @ times_times_assn @ one_one_assn @ Zs ) )
      = ( foldl_assn_assn @ times_times_assn @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4705_foldl__absorb1,axiom,
    ! [X4: complex,Zs: list_complex] :
      ( ( times_times_complex @ X4 @ ( foldl_2274445284955862271omplex @ times_times_complex @ one_one_complex @ Zs ) )
      = ( foldl_2274445284955862271omplex @ times_times_complex @ X4 @ Zs ) ) ).

% foldl_absorb1
thf(fact_4706_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > nat > assn,A: nat,As: list_nat,C: nat,Cs: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ ( cons_nat @ A @ As ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L8301102511889123557at_nat @ P @ As @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4707_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > int > assn,A: nat,As: list_nat,C: int,Cs: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ ( cons_nat @ A @ As ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L8298612041380073281at_int @ P @ As @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4708_list__assn_Osimps_I2_J,axiom,
    ! [P: int > nat > assn,A: int,As: list_int,C: nat,Cs: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ ( cons_int @ A @ As ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L77084186935402305nt_nat @ P @ As @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4709_list__assn_Osimps_I2_J,axiom,
    ! [P: int > int > assn,A: int,As: list_int,C: int,Cs: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ ( cons_int @ A @ As ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L74593716426352029nt_int @ P @ As @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4710_list__assn_Osimps_I2_J,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] :
      ( ( vEBT_L6296928887356842470_VEBTi @ P @ ( cons_VEBT_VEBT @ A @ As ) @ ( cons_VEBT_VEBTi @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L6296928887356842470_VEBTi @ P @ As @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4711_list__assn__simps_I2_J,axiom,
    ! [P: nat > nat > assn,A: nat,As: list_nat,C: nat,Cs: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ ( cons_nat @ A @ As ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L8301102511889123557at_nat @ P @ As @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4712_list__assn__simps_I2_J,axiom,
    ! [P: nat > int > assn,A: nat,As: list_nat,C: int,Cs: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ ( cons_nat @ A @ As ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L8298612041380073281at_int @ P @ As @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4713_list__assn__simps_I2_J,axiom,
    ! [P: int > nat > assn,A: int,As: list_int,C: nat,Cs: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ ( cons_int @ A @ As ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L77084186935402305nt_nat @ P @ As @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4714_list__assn__simps_I2_J,axiom,
    ! [P: int > int > assn,A: int,As: list_int,C: int,Cs: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ ( cons_int @ A @ As ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L74593716426352029nt_int @ P @ As @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4715_list__assn__simps_I2_J,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,A: vEBT_VEBT,As: list_VEBT_VEBT,C: vEBT_VEBTi,Cs: list_VEBT_VEBTi] :
      ( ( vEBT_L6296928887356842470_VEBTi @ P @ ( cons_VEBT_VEBT @ A @ As ) @ ( cons_VEBT_VEBTi @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A @ C ) @ ( vEBT_L6296928887356842470_VEBTi @ P @ As @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4716_ex__less__of__nat__mult,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4717_ex__less__of__nat__mult,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4718_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri8010041392384452111omplex @ K )
       != zero_zero_complex )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4719_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_4720_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_4721_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_4722_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_4723_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4724_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_4725_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_4726_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_4727_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_real @ Xs2 ) )
      = ( ? [X: real,Ys3: list_real] :
            ( ( Xs2
              = ( cons_real @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_real @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4728_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs2 ) )
      = ( ? [X: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4729_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
      = ( ? [X: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4730_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs2 ) )
      = ( ? [X: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4731_sorted__simps_I2_J,axiom,
    ! [X4: real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ ( cons_real @ X4 @ Ys ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Ys ) )
           => ( ord_less_eq_real @ X4 @ X ) )
        & ( sorted_wrt_real @ ord_less_eq_real @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4732_sorted__simps_I2_J,axiom,
    ! [X4: $o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( cons_o @ X4 @ Ys ) )
      = ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Ys ) )
           => ( ord_less_eq_o @ X4 @ X ) )
        & ( sorted_wrt_o @ ord_less_eq_o @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4733_sorted__simps_I2_J,axiom,
    ! [X4: rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ X4 @ Ys ) )
      = ( ! [X: rat] :
            ( ( member_rat @ X @ ( set_rat2 @ Ys ) )
           => ( ord_less_eq_rat @ X4 @ X ) )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4734_sorted__simps_I2_J,axiom,
    ! [X4: num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ X4 @ Ys ) )
      = ( ! [X: num] :
            ( ( member_num @ X @ ( set_num2 @ Ys ) )
           => ( ord_less_eq_num @ X4 @ X ) )
        & ( sorted_wrt_num @ ord_less_eq_num @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4735_sorted__simps_I2_J,axiom,
    ! [X4: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X4 @ Ys ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ( ord_less_eq_nat @ X4 @ X ) )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4736_sorted__simps_I2_J,axiom,
    ! [X4: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X4 @ Ys ) )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Ys ) )
           => ( ord_less_eq_int @ X4 @ X ) )
        & ( sorted_wrt_int @ ord_less_eq_int @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4737_strict__sorted__simps_I2_J,axiom,
    ! [X4: $o,Ys: list_o] :
      ( ( sorted_wrt_o @ ord_less_o @ ( cons_o @ X4 @ Ys ) )
      = ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Ys ) )
           => ( ord_less_o @ X4 @ X ) )
        & ( sorted_wrt_o @ ord_less_o @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4738_strict__sorted__simps_I2_J,axiom,
    ! [X4: real,Ys: list_real] :
      ( ( sorted_wrt_real @ ord_less_real @ ( cons_real @ X4 @ Ys ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Ys ) )
           => ( ord_less_real @ X4 @ X ) )
        & ( sorted_wrt_real @ ord_less_real @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4739_strict__sorted__simps_I2_J,axiom,
    ! [X4: rat,Ys: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ ( cons_rat @ X4 @ Ys ) )
      = ( ! [X: rat] :
            ( ( member_rat @ X @ ( set_rat2 @ Ys ) )
           => ( ord_less_rat @ X4 @ X ) )
        & ( sorted_wrt_rat @ ord_less_rat @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4740_strict__sorted__simps_I2_J,axiom,
    ! [X4: num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ ( cons_num @ X4 @ Ys ) )
      = ( ! [X: num] :
            ( ( member_num @ X @ ( set_num2 @ Ys ) )
           => ( ord_less_num @ X4 @ X ) )
        & ( sorted_wrt_num @ ord_less_num @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4741_strict__sorted__simps_I2_J,axiom,
    ! [X4: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X4 @ Ys ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ( ord_less_nat @ X4 @ X ) )
        & ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4742_strict__sorted__simps_I2_J,axiom,
    ! [X4: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ ( cons_int @ X4 @ Ys ) )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Ys ) )
           => ( ord_less_int @ X4 @ X ) )
        & ( sorted_wrt_int @ ord_less_int @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4743_TBOUND__of__list,axiom,
    ! [Xs2: list_o] : ( time_TBOUND_array_o @ ( array_of_list_o @ Xs2 ) @ ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_4744_TBOUND__of__list,axiom,
    ! [Xs2: list_nat] : ( time_T3621710982123377501ay_nat @ ( array_of_list_nat @ Xs2 ) @ ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_4745_TBOUND__of__list,axiom,
    ! [Xs2: list_int] : ( time_T8667231999468956601ay_int @ ( array_of_list_int @ Xs2 ) @ ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_4746_foldl__length__aux,axiom,
    ! [A: nat,L: list_real] :
      ( ( foldl_nat_real
        @ ^ [I3: nat,X: real] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldl_length_aux
thf(fact_4747_foldl__length__aux,axiom,
    ! [A: nat,L: list_o] :
      ( ( foldl_nat_o
        @ ^ [I3: nat,X: $o] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldl_length_aux
thf(fact_4748_foldl__length__aux,axiom,
    ! [A: nat,L: list_nat] :
      ( ( foldl_nat_nat
        @ ^ [I3: nat,X: nat] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldl_length_aux
thf(fact_4749_foldl__length__aux,axiom,
    ! [A: nat,L: list_int] :
      ( ( foldl_nat_int
        @ ^ [I3: nat,X: int] : ( suc @ I3 )
        @ A
        @ L )
      = ( plus_plus_nat @ A @ ( size_size_list_int @ L ) ) ) ).

% foldl_length_aux
thf(fact_4750_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_4751_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_4752_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_4753_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_4754_nth__Cons_H,axiom,
    ! [N: nat,X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ N )
          = X4 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ N )
          = ( nth_VEBT_VEBTi @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4755_nth__Cons_H,axiom,
    ! [N: nat,X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ N )
          = X4 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ N )
          = ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4756_nth__Cons_H,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ N )
          = X4 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ N )
          = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4757_nth__Cons_H,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ N )
          = X4 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ N )
          = ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4758_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( n_lists_nat @ ( suc @ N ) @ Xs2 )
      = ( concat_list_nat
        @ ( map_li960784813134754710st_nat
          @ ^ [Ys3: list_nat] :
              ( map_nat_list_nat
              @ ^ [Y4: nat] : ( cons_nat @ Y4 @ Ys3 )
              @ Xs2 )
          @ ( n_lists_nat @ N @ Xs2 ) ) ) ) ).

% n_lists.simps(2)
thf(fact_4759_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( n_lists_int @ ( suc @ N ) @ Xs2 )
      = ( concat_list_int
        @ ( map_li8902190837986183758st_int
          @ ^ [Ys3: list_int] :
              ( map_int_list_int
              @ ^ [Y4: int] : ( cons_int @ Y4 @ Ys3 )
              @ Xs2 )
          @ ( n_lists_int @ N @ Xs2 ) ) ) ) ).

% n_lists.simps(2)
thf(fact_4760_Id__on__fold,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( id_on_complex @ A3 )
        = ( finite908486625158157349omplex
          @ ^ [X: complex] : ( insert3126710022685806477omplex @ ( produc101793102246108661omplex @ X @ X ) )
          @ bot_bo313579480278226313omplex
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_4761_Id__on__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( id_on_nat @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) )
          @ bot_bo2099793752762293965at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_4762_Id__on__fold,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( id_on_int @ A3 )
        = ( finite5202366122487795491nt_int
          @ ^ [X: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) )
          @ bot_bo1796632182523588997nt_int
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_4763_Id__on__fold,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( id_on_Code_integer @ A3 )
        = ( finite6473783728727339668nteger
          @ ^ [X: code_integer] : ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ X @ X ) )
          @ bot_bo4276436098303576167nteger
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_4764_list_Osize__gen_I2_J,axiom,
    ! [X4: nat > nat,X21: nat,X22: list_nat] :
      ( ( size_list_nat @ X4 @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( X4 @ X21 ) @ ( size_list_nat @ X4 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size_gen(2)
thf(fact_4765_list_Osize__gen_I2_J,axiom,
    ! [X4: int > nat,X21: int,X22: list_int] :
      ( ( size_list_int @ X4 @ ( cons_int @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( X4 @ X21 ) @ ( size_list_int @ X4 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size_gen(2)
thf(fact_4766_list_Osize__gen_I2_J,axiom,
    ! [X4: vEBT_VEBT > nat,X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
      ( ( size_list_VEBT_VEBT @ X4 @ ( cons_VEBT_VEBT @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( X4 @ X21 ) @ ( size_list_VEBT_VEBT @ X4 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size_gen(2)
thf(fact_4767_nth__non__equal__first__eq,axiom,
    ! [X4: vEBT_VEBTi,Y: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,N: nat] :
      ( ( X4 != Y )
     => ( ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBTi @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4768_nth__non__equal__first__eq,axiom,
    ! [X4: vEBT_VEBT,Y: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( X4 != Y )
     => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4769_nth__non__equal__first__eq,axiom,
    ! [X4: nat,Y: nat,Xs2: list_nat,N: nat] :
      ( ( X4 != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4770_nth__non__equal__first__eq,axiom,
    ! [X4: int,Y: int,Xs2: list_int,N: nat] :
      ( ( X4 != Y )
     => ( ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4771_nth__equal__first__eq,axiom,
    ! [X4: vEBT_VEBTi,Xs2: list_VEBT_VEBTi,N: nat] :
      ( ~ ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( ( nth_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4772_nth__equal__first__eq,axiom,
    ! [X4: set_nat,Xs2: list_set_nat,N: nat] :
      ( ~ ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( ( ( nth_set_nat @ ( cons_set_nat @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4773_nth__equal__first__eq,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ~ ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4774_nth__equal__first__eq,axiom,
    ! [X4: real,Xs2: list_real,N: nat] :
      ( ~ ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( ( nth_real @ ( cons_real @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4775_nth__equal__first__eq,axiom,
    ! [X4: $o,Xs2: list_o,N: nat] :
      ( ~ ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( ( nth_o @ ( cons_o @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4776_nth__equal__first__eq,axiom,
    ! [X4: nat,Xs2: list_nat,N: nat] :
      ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ( nth_nat @ ( cons_nat @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4777_nth__equal__first__eq,axiom,
    ! [X4: int,Xs2: list_int,N: nat] :
      ( ~ ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( ( nth_int @ ( cons_int @ X4 @ Xs2 ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4778_Cons__replicate__eq,axiom,
    ! [X4: nat,Xs2: list_nat,N: nat,Y: nat] :
      ( ( ( cons_nat @ X4 @ Xs2 )
        = ( replicate_nat @ N @ Y ) )
      = ( ( X4 = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X4 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_4779_Cons__replicate__eq,axiom,
    ! [X4: int,Xs2: list_int,N: nat,Y: int] :
      ( ( ( cons_int @ X4 @ Xs2 )
        = ( replicate_int @ N @ Y ) )
      = ( ( X4 = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_int @ ( minus_minus_nat @ N @ one_one_nat ) @ X4 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_4780_Cons__replicate__eq,axiom,
    ! [X4: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( cons_VEBT_VEBT @ X4 @ Xs2 )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( X4 = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_VEBT_VEBT @ ( minus_minus_nat @ N @ one_one_nat ) @ X4 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_4781_Cons__replicate__eq,axiom,
    ! [X4: $o,Xs2: list_o,N: nat,Y: $o] :
      ( ( ( cons_o @ X4 @ Xs2 )
        = ( replicate_o @ N @ Y ) )
      = ( ( X4 = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_o @ ( minus_minus_nat @ N @ one_one_nat ) @ X4 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_4782_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X4: nat,Xs2: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X4 @ Xs2 ) )
          = ( cons_nat @ X4 @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs2 ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X4 @ Xs2 ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs2 ) ) ) ) ).

% slice_Cons
thf(fact_4783_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X4: int,Xs2: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X4 @ Xs2 ) )
          = ( cons_int @ X4 @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs2 ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X4 @ Xs2 ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs2 ) ) ) ) ).

% slice_Cons
thf(fact_4784_time__array__of__list,axiom,
    ! [Xs2: list_o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_o @ ( array_of_list_o @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_4785_time__array__of__list,axiom,
    ! [Xs2: list_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_nat @ ( array_of_list_nat @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_4786_time__array__of__list,axiom,
    ! [Xs2: list_int,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_int @ ( array_of_list_int @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_4787_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_4788_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_4789_not__real__square__gt__zero,axiom,
    ! [X4: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
      = ( X4 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_4790_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N4: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N4
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_4791_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N4: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N4
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_4792_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N4: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N4
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_4793_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N4
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_4794_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N4
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_4795_length__Cons,axiom,
    ! [X4: real,Xs2: list_real] :
      ( ( size_size_list_real @ ( cons_real @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_Cons
thf(fact_4796_length__Cons,axiom,
    ! [X4: $o,Xs2: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_Cons
thf(fact_4797_length__Cons,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_Cons
thf(fact_4798_length__Cons,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X4 @ Xs2 ) )
      = ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_Cons
thf(fact_4799_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A2: rat,N4: nat] :
          ( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_4800_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A2: complex,N4: nat] :
          ( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A2 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_4801_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A2: real,N4: nat] :
          ( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A2 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_4802_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A2: int,N4: nat] :
          ( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A2 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_4803_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A2: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A2 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_4804_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X4 )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( 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_4805_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_4806_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_4807_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_4808_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_4809_pochhammer__of__nat,axiom,
    ! [X4: nat,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X4 ) @ N )
      = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4810_pochhammer__of__nat,axiom,
    ! [X4: nat,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X4 ) @ N )
      = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4811_pochhammer__of__nat,axiom,
    ! [X4: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N )
      = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_4812_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z2: int] :
        ? [N4: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_4813_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_4814_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4815_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_4816_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4817_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4818_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_4819_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_4820_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_4821_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_4822_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4823_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_4824_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_4825_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_4826_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_4827_pochhammer__pos,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_4828_pochhammer__pos,axiom,
    ! [X4: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_4829_pochhammer__pos,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_4830_pochhammer__pos,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).

% pochhammer_pos
thf(fact_4831_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ M )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4832_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4833_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4834_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ N )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4835_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4836_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4837_pochhammer__nonneg,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4838_pochhammer__nonneg,axiom,
    ! [X4: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4839_pochhammer__nonneg,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4840_pochhammer__nonneg,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_4841_zdiff__int__split,axiom,
    ! [P: int > $o,X4: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X4 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X4 @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_4842_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_4843_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_4844_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_4845_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_4846_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_4847_pochhammer__rec,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4848_pochhammer__rec,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4849_pochhammer__rec,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4850_pochhammer__rec,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4851_pochhammer__rec,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_4852_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4853_pochhammer__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4854_pochhammer__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4855_pochhammer__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4856_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_4857_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4858_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4859_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4860_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4861_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_4862_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4863_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4864_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4865_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4866_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4867_subset__Collect__iff,axiom,
    ! [B4: set_VEBT_VEBT,A3: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_le4337996190870823476T_VEBT @ B4 @ A3 )
     => ( ( ord_le4337996190870823476T_VEBT @ B4
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4868_subset__Collect__iff,axiom,
    ! [B4: set_real,A3: set_real,P: real > $o] :
      ( ( ord_less_eq_set_real @ B4 @ A3 )
     => ( ( ord_less_eq_set_real @ B4
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: real] :
              ( ( member_real @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4869_subset__Collect__iff,axiom,
    ! [B4: set_complex,A3: set_complex,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ B4 @ A3 )
     => ( ( ord_le211207098394363844omplex @ B4
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4870_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A3: set_list_nat,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
     => ( ( ord_le6045566169113846134st_nat @ B4
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: list_nat] :
              ( ( member_list_nat @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4871_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A3: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ B4
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: set_nat] :
              ( ( member_set_nat @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4872_subset__Collect__iff,axiom,
    ! [B4: set_nat,A3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A3 )
     => ( ( ord_less_eq_set_nat @ B4
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4873_subset__Collect__iff,axiom,
    ! [B4: set_int,A3: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ B4 @ A3 )
     => ( ( ord_less_eq_set_int @ B4
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ B4 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_4874_subset__CollectI,axiom,
    ! [B4: set_VEBT_VEBT,A3: set_VEBT_VEBT,Q: vEBT_VEBT > $o,P: vEBT_VEBT > $o] :
      ( ( ord_le4337996190870823476T_VEBT @ B4 @ A3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le4337996190870823476T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4875_subset__CollectI,axiom,
    ! [B4: set_real,A3: set_real,Q: real > $o,P: real > $o] :
      ( ( ord_less_eq_set_real @ B4 @ A3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4876_subset__CollectI,axiom,
    ! [B4: set_complex,A3: set_complex,Q: complex > $o,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ B4 @ A3 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4877_subset__CollectI,axiom,
    ! [B4: set_list_nat,A3: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4878_subset__CollectI,axiom,
    ! [B4: set_set_nat,A3: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ A3 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4879_subset__CollectI,axiom,
    ! [B4: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4880_subset__CollectI,axiom,
    ! [B4: set_int,A3: set_int,Q: int > $o,P: int > $o] :
      ( ( ord_less_eq_set_int @ B4 @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ B4 )
                & ( Q @ X ) ) )
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_4881_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4882_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s2602460028002588243omplex @ Z @ N )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4883_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s7457072308508201937r_real @ Z @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4884_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4885_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4886_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_4887_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( 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_4888_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) )
      = ( vEBT_V9176841429113362141ildupi @ N ) ) ).

% Tbuildupi_buildupi'
thf(fact_4889_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A4: real,B3: real,C4: real] :
            ( ( P @ A4 @ B3 )
           => ( ( P @ B3 @ C4 )
             => ( ( ord_less_eq_real @ A4 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C4 )
                 => ( P @ A4 @ C4 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A4: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A4 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D3 ) )
                       => ( P @ A4 @ B3 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_4890_Set__filter__fold,axiom,
    ! [A3: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( filter_VEBT_VEBT @ P @ A3 )
        = ( finite2755337914281901096T_VEBT
          @ ^ [X: vEBT_VEBT,A9: set_VEBT_VEBT] : ( if_set_VEBT_VEBT @ ( P @ X ) @ ( insert_VEBT_VEBT @ X @ A9 ) @ A9 )
          @ bot_bo8194388402131092736T_VEBT
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4891_Set__filter__fold,axiom,
    ! [A3: set_complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( filter_complex @ P @ A3 )
        = ( finite7687009701937157724omplex
          @ ^ [X: complex,A9: set_complex] : ( if_set_complex @ ( P @ X ) @ ( insert_complex @ X @ A9 ) @ A9 )
          @ bot_bot_set_complex
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4892_Set__filter__fold,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( filter_Code_integer @ P @ A3 )
        = ( finite7486036041278456378nteger
          @ ^ [X: code_integer,A9: set_Code_integer] : ( if_set_Code_integer @ ( P @ X ) @ ( insert_Code_integer @ X @ A9 ) @ A9 )
          @ bot_bo3990330152332043303nteger
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4893_Set__filter__fold,axiom,
    ! [A3: set_real,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( filter_real @ P @ A3 )
        = ( finite3596656892938901080t_real
          @ ^ [X: real,A9: set_real] : ( if_set_real @ ( P @ X ) @ ( insert_real @ X @ A9 ) @ A9 )
          @ bot_bot_set_real
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4894_Set__filter__fold,axiom,
    ! [A3: set_o,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( filter_o @ P @ A3 )
        = ( finite_fold_o_set_o
          @ ^ [X: $o,A9: set_o] : ( if_set_o @ ( P @ X ) @ ( insert_o @ X @ A9 ) @ A9 )
          @ bot_bot_set_o
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4895_Set__filter__fold,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( filter_nat2 @ P @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [X: nat,A9: set_nat] : ( if_set_nat @ ( P @ X ) @ ( insert_nat @ X @ A9 ) @ A9 )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4896_Set__filter__fold,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( filter_int @ P @ A3 )
        = ( finite1723285575846220376et_int
          @ ^ [X: int,A9: set_int] : ( if_set_int @ ( P @ X ) @ ( insert_int @ X @ A9 ) @ A9 )
          @ bot_bot_set_int
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_4897_graph__map__upd,axiom,
    ! [M: int > option_int,K: int,V: int] :
      ( ( graph_int_int @ ( fun_up8666045135305973159on_int @ M @ K @ ( some_int @ V ) ) )
      = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ ( fun_up8666045135305973159on_int @ M @ K @ none_int ) ) ) ) ).

% graph_map_upd
thf(fact_4898_graph__map__upd,axiom,
    ! [M: code_integer > option_Code_integer,K: code_integer,V: code_integer] :
      ( ( graph_5282091004195177018nteger @ ( fun_up6566892301432185865nteger @ M @ K @ ( some_Code_integer @ V ) ) )
      = ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ K @ V ) @ ( graph_5282091004195177018nteger @ ( fun_up6566892301432185865nteger @ M @ K @ none_Code_integer ) ) ) ) ).

% graph_map_upd
thf(fact_4899_graph__map__upd,axiom,
    ! [M: nat > option_nat,K: nat,V: nat] :
      ( ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ ( some_nat @ V ) ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ none_nat ) ) ) ) ).

% graph_map_upd
thf(fact_4900_graph__map__upd,axiom,
    ! [M: vEBT_VEBT > option_nat,K: vEBT_VEBT,V: nat] :
      ( ( graph_VEBT_VEBT_nat @ ( fun_up5885881570350532375on_nat @ M @ K @ ( some_nat @ V ) ) )
      = ( insert8978894354669351395BT_nat @ ( produc738532404422230701BT_nat @ K @ V ) @ ( graph_VEBT_VEBT_nat @ ( fun_up5885881570350532375on_nat @ M @ K @ none_nat ) ) ) ) ).

% graph_map_upd
thf(fact_4901_graph__map__upd,axiom,
    ! [M: nat > option_num,K: nat,V: num] :
      ( ( graph_nat_num @ ( fun_up2201401324907169337on_num @ M @ K @ ( some_num @ V ) ) )
      = ( insert8920054152555992091at_num @ ( product_Pair_nat_num @ K @ V ) @ ( graph_nat_num @ ( fun_up2201401324907169337on_num @ M @ K @ none_num ) ) ) ) ).

% graph_map_upd
thf(fact_4902_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_4903_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_4904_sup__Some,axiom,
    ! [X4: assn,Y: assn] :
      ( ( sup_sup_option_assn @ ( some_assn @ X4 ) @ ( some_assn @ Y ) )
      = ( some_assn @ ( sup_sup_assn @ X4 @ Y ) ) ) ).

% sup_Some
thf(fact_4905_sup__Some,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( sup_su3598758113090618626et_nat @ ( some_set_nat @ X4 ) @ ( some_set_nat @ Y ) )
      = ( some_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) ) ) ).

% sup_Some
thf(fact_4906_sup__Some,axiom,
    ! [X4: nat,Y: nat] :
      ( ( sup_sup_option_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
      = ( some_nat @ ( sup_sup_nat @ X4 @ Y ) ) ) ).

% sup_Some
thf(fact_4907_Un__empty,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ( sup_sup_set_real @ A3 @ B4 )
        = bot_bot_set_real )
      = ( ( A3 = bot_bot_set_real )
        & ( B4 = bot_bot_set_real ) ) ) ).

% Un_empty
thf(fact_4908_Un__empty,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( ( sup_sup_set_o @ A3 @ B4 )
        = bot_bot_set_o )
      = ( ( A3 = bot_bot_set_o )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_4909_Un__empty,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B4 )
        = bot_bot_set_nat )
      = ( ( A3 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_4910_Un__empty,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ( sup_sup_set_int @ A3 @ B4 )
        = bot_bot_set_int )
      = ( ( A3 = bot_bot_set_int )
        & ( B4 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_4911_Un__subset__iff,axiom,
    ! [A3: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B4 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A3 @ C2 )
        & ( ord_less_eq_set_nat @ B4 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_4912_Un__subset__iff,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B4 ) @ C2 )
      = ( ( ord_less_eq_set_int @ A3 @ C2 )
        & ( ord_less_eq_set_int @ B4 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_4913_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_4914_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_4915_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_4916_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_4917_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z2: int] :
        ? [N4: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_4918_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_4919_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_4920_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_4921_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_4922_pos__mult__pos__ge,axiom,
    ! [X4: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X4 ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_4923_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R5: int,Q5: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( ord_less_int @ B @ R5 )
           => ( ord_less_eq_int @ Q5 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_4924_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R5: int,Q5: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
       => ( ( ord_less_int @ R5 @ B )
         => ( ( ord_less_int @ R3 @ B )
           => ( ord_less_eq_int @ Q6 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_4925_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q5: int,R3: int,B6: int,Q6: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R5 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R5 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
           => ( ( ord_less_int @ zero_zero_int @ B6 )
             => ( ( ord_less_eq_int @ B6 @ B )
               => ( ord_less_eq_int @ Q6 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_4926_zdiv__mono2__lemma,axiom,
    ! [B: int,Q5: int,R3: int,B6: int,Q6: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R5 ) )
       => ( ( ord_less_int @ R5 @ B6 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B6 )
             => ( ( ord_less_eq_int @ B6 @ B )
               => ( ord_less_eq_int @ Q5 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_4927_q__pos__lemma,axiom,
    ! [B6: int,Q6: int,R5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R5 ) )
     => ( ( ord_less_int @ R5 @ B6 )
       => ( ( ord_less_int @ zero_zero_int @ B6 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_4928_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_4929_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_4930_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_4931_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_4932_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_4933_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_4934_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4935_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_4936_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_4937_plusinfinity,axiom,
    ! [D: int,P4: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P4 @ X3 )
            = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P4 @ X3 ) ) )
         => ( ? [X_1: int] : ( P4 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_4938_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_4939_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_4940_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_4941_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_4942_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_4943_Collect__disj__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X: complex] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_complex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4944_Collect__disj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4945_Collect__disj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4946_Collect__disj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4947_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4948_Un__def,axiom,
    ( sup_su6272177626956685416T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A5 )
              | ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4949_Un__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A5 )
              | ( member_real @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4950_Un__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A5 )
              | ( member_complex @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4951_Un__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A5 )
              | ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4952_Un__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A5 )
              | ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4953_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A5 )
              | ( member_int @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4954_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              | ( member_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_4955_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_4956_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_4957_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_4958_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_4959_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_4960_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_4961_imp__le__cong,axiom,
    ! [X4: int,X9: int,P: $o,P4: $o] :
      ( ( X4 = X9 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X9 )
           => P4 ) ) ) ) ).

% imp_le_cong
thf(fact_4962_conj__le__cong,axiom,
    ! [X4: int,X9: int,P: $o,P4: $o] :
      ( ( X4 = X9 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X9 )
            & P4 ) ) ) ) ).

% conj_le_cong
thf(fact_4963_verit__la__generic,axiom,
    ! [A: int,X4: int] :
      ( ( ord_less_eq_int @ A @ X4 )
      | ( A = X4 )
      | ( ord_less_eq_int @ X4 @ A ) ) ).

% verit_la_generic
thf(fact_4964_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_4965_boolean__algebra_Odisj__zero__right,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ X4 @ bot_bot_assn )
      = X4 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4966_boolean__algebra_Odisj__zero__right,axiom,
    ! [X4: set_real] :
      ( ( sup_sup_set_real @ X4 @ bot_bot_set_real )
      = X4 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4967_boolean__algebra_Odisj__zero__right,axiom,
    ! [X4: set_o] :
      ( ( sup_sup_set_o @ X4 @ bot_bot_set_o )
      = X4 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4968_boolean__algebra_Odisj__zero__right,axiom,
    ! [X4: set_nat] :
      ( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
      = X4 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4969_boolean__algebra_Odisj__zero__right,axiom,
    ! [X4: set_int] :
      ( ( sup_sup_set_int @ X4 @ bot_bot_set_int )
      = X4 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4970_foldl__un__empty__eq,axiom,
    ! [I: set_real,Ww: list_set_real] :
      ( ( foldl_4142838678319929319t_real @ sup_sup_set_real @ I @ Ww )
      = ( sup_sup_set_real @ I @ ( foldl_4142838678319929319t_real @ sup_sup_set_real @ bot_bot_set_real @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_4971_foldl__un__empty__eq,axiom,
    ! [I: set_o,Ww: list_set_o] :
      ( ( foldl_set_o_set_o @ sup_sup_set_o @ I @ Ww )
      = ( sup_sup_set_o @ I @ ( foldl_set_o_set_o @ sup_sup_set_o @ bot_bot_set_o @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_4972_foldl__un__empty__eq,axiom,
    ! [I: set_nat,Ww: list_set_nat] :
      ( ( foldl_4988731653086973103et_nat @ sup_sup_set_nat @ I @ Ww )
      = ( sup_sup_set_nat @ I @ ( foldl_4988731653086973103et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_4973_foldl__un__empty__eq,axiom,
    ! [I: set_int,Ww: list_set_int] :
      ( ( foldl_6819690284573351271et_int @ sup_sup_set_int @ I @ Ww )
      = ( sup_sup_set_int @ I @ ( foldl_6819690284573351271et_int @ sup_sup_set_int @ bot_bot_set_int @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_4974_Un__empty__left,axiom,
    ! [B4: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_4975_Un__empty__left,axiom,
    ! [B4: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_4976_Un__empty__left,axiom,
    ! [B4: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_4977_Un__empty__left,axiom,
    ! [B4: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_4978_Un__empty__right,axiom,
    ! [A3: set_real] :
      ( ( sup_sup_set_real @ A3 @ bot_bot_set_real )
      = A3 ) ).

% Un_empty_right
thf(fact_4979_Un__empty__right,axiom,
    ! [A3: set_o] :
      ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Un_empty_right
thf(fact_4980_Un__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_4981_Un__empty__right,axiom,
    ! [A3: set_int] :
      ( ( sup_sup_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Un_empty_right
thf(fact_4982_Un__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B4: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B4 @ D4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B4 ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_4983_Un__mono,axiom,
    ! [A3: set_int,C2: set_int,B4: set_int,D4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ B4 @ D4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B4 ) @ ( sup_sup_set_int @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_4984_Un__least,axiom,
    ! [A3: set_nat,C2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B4 ) @ C2 ) ) ) ).

% Un_least
thf(fact_4985_Un__least,axiom,
    ! [A3: set_int,C2: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ B4 @ C2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B4 ) @ C2 ) ) ) ).

% Un_least
thf(fact_4986_Un__upper1,axiom,
    ! [A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B4 ) ) ).

% Un_upper1
thf(fact_4987_Un__upper1,axiom,
    ! [A3: set_int,B4: set_int] : ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ A3 @ B4 ) ) ).

% Un_upper1
thf(fact_4988_Un__upper2,axiom,
    ! [B4: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A3 @ B4 ) ) ).

% Un_upper2
thf(fact_4989_Un__upper2,axiom,
    ! [B4: set_int,A3: set_int] : ( ord_less_eq_set_int @ B4 @ ( sup_sup_set_int @ A3 @ B4 ) ) ).

% Un_upper2
thf(fact_4990_Un__absorb1,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( sup_sup_set_nat @ A3 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_4991_Un__absorb1,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( sup_sup_set_int @ A3 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_4992_Un__absorb2,axiom,
    ! [B4: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A3 )
     => ( ( sup_sup_set_nat @ A3 @ B4 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_4993_Un__absorb2,axiom,
    ! [B4: set_int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B4 @ A3 )
     => ( ( sup_sup_set_int @ A3 @ B4 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_4994_subset__UnE,axiom,
    ! [C2: set_nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B4 ) )
     => ~ ! [A10: set_nat] :
            ( ( ord_less_eq_set_nat @ A10 @ A3 )
           => ! [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ B7 @ B4 )
               => ( C2
                 != ( sup_sup_set_nat @ A10 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_4995_subset__UnE,axiom,
    ! [C2: set_int,A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A3 @ B4 ) )
     => ~ ! [A10: set_int] :
            ( ( ord_less_eq_set_int @ A10 @ A3 )
           => ! [B7: set_int] :
                ( ( ord_less_eq_set_int @ B7 @ B4 )
               => ( C2
                 != ( sup_sup_set_int @ A10 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_4996_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_4997_subset__Un__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( sup_sup_set_int @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_4998_Set_Ofilter__def,axiom,
    ( filter_VEBT_VEBT
    = ( ^ [P2: vEBT_VEBT > $o,A5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [A2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_4999_Set_Ofilter__def,axiom,
    ( filter_real
    = ( ^ [P2: real > $o,A5: set_real] :
          ( collect_real
          @ ^ [A2: real] :
              ( ( member_real @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5000_Set_Ofilter__def,axiom,
    ( filter_complex
    = ( ^ [P2: complex > $o,A5: set_complex] :
          ( collect_complex
          @ ^ [A2: complex] :
              ( ( member_complex @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5001_Set_Ofilter__def,axiom,
    ( filter_list_nat
    = ( ^ [P2: list_nat > $o,A5: set_list_nat] :
          ( collect_list_nat
          @ ^ [A2: list_nat] :
              ( ( member_list_nat @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5002_Set_Ofilter__def,axiom,
    ( filter_set_nat
    = ( ^ [P2: set_nat > $o,A5: set_set_nat] :
          ( collect_set_nat
          @ ^ [A2: set_nat] :
              ( ( member_set_nat @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5003_Set_Ofilter__def,axiom,
    ( filter_nat2
    = ( ^ [P2: nat > $o,A5: set_nat] :
          ( collect_nat
          @ ^ [A2: nat] :
              ( ( member_nat @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5004_Set_Ofilter__def,axiom,
    ( filter_int
    = ( ^ [P2: int > $o,A5: set_int] :
          ( collect_int
          @ ^ [A2: int] :
              ( ( member_int @ A2 @ A5 )
              & ( P2 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_5005_insert__def,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A2: vEBT_VEBT] :
          ( sup_su6272177626956685416T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5006_insert__def,axiom,
    ( insert_o
    = ( ^ [A2: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X: $o] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5007_insert__def,axiom,
    ( insert_real
    = ( ^ [A2: real] :
          ( sup_sup_set_real
          @ ( collect_real
            @ ^ [X: real] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5008_insert__def,axiom,
    ( insert_complex
    = ( ^ [A2: complex] :
          ( sup_sup_set_complex
          @ ( collect_complex
            @ ^ [X: complex] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5009_insert__def,axiom,
    ( insert_list_nat
    = ( ^ [A2: list_nat] :
          ( sup_sup_set_list_nat
          @ ( collect_list_nat
            @ ^ [X: list_nat] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5010_insert__def,axiom,
    ( insert_set_nat
    = ( ^ [A2: set_nat] :
          ( sup_sup_set_set_nat
          @ ( collect_set_nat
            @ ^ [X: set_nat] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5011_insert__def,axiom,
    ( insert_int
    = ( ^ [A2: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X: int] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5012_insert__def,axiom,
    ( insert_nat
    = ( ^ [A2: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_5013_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_5014_insert__is__Un,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A2: vEBT_VEBT] : ( sup_su6272177626956685416T_VEBT @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% insert_is_Un
thf(fact_5015_insert__is__Un,axiom,
    ( insert_real
    = ( ^ [A2: real] : ( sup_sup_set_real @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% insert_is_Un
thf(fact_5016_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A2: $o] : ( sup_sup_set_o @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_5017_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_5018_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A2: int] : ( sup_sup_set_int @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_5019_Un__singleton__iff,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ( sup_su6272177626956685416T_VEBT @ A3 @ B4 )
        = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( ( A3 = bot_bo8194388402131092736T_VEBT )
          & ( B4
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
        | ( ( A3
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
          & ( B4 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A3
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
          & ( B4
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5020_Un__singleton__iff,axiom,
    ! [A3: set_real,B4: set_real,X4: real] :
      ( ( ( sup_sup_set_real @ A3 @ B4 )
        = ( insert_real @ X4 @ bot_bot_set_real ) )
      = ( ( ( A3 = bot_bot_set_real )
          & ( B4
            = ( insert_real @ X4 @ bot_bot_set_real ) ) )
        | ( ( A3
            = ( insert_real @ X4 @ bot_bot_set_real ) )
          & ( B4 = bot_bot_set_real ) )
        | ( ( A3
            = ( insert_real @ X4 @ bot_bot_set_real ) )
          & ( B4
            = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5021_Un__singleton__iff,axiom,
    ! [A3: set_o,B4: set_o,X4: $o] :
      ( ( ( sup_sup_set_o @ A3 @ B4 )
        = ( insert_o @ X4 @ bot_bot_set_o ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B4
            = ( insert_o @ X4 @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X4 @ bot_bot_set_o ) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X4 @ bot_bot_set_o ) )
          & ( B4
            = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5022_Un__singleton__iff,axiom,
    ! [A3: set_nat,B4: set_nat,X4: nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B4 )
        = ( insert_nat @ X4 @ bot_bot_set_nat ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X4 @ bot_bot_set_nat ) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X4 @ bot_bot_set_nat ) )
          & ( B4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5023_Un__singleton__iff,axiom,
    ! [A3: set_int,B4: set_int,X4: int] :
      ( ( ( sup_sup_set_int @ A3 @ B4 )
        = ( insert_int @ X4 @ bot_bot_set_int ) )
      = ( ( ( A3 = bot_bot_set_int )
          & ( B4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) )
        | ( ( A3
            = ( insert_int @ X4 @ bot_bot_set_int ) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A3
            = ( insert_int @ X4 @ bot_bot_set_int ) )
          & ( B4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5024_singleton__Un__iff,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT )
        = ( sup_su6272177626956685416T_VEBT @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bo8194388402131092736T_VEBT )
          & ( B4
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
        | ( ( A3
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
          & ( B4 = bot_bo8194388402131092736T_VEBT ) )
        | ( ( A3
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
          & ( B4
            = ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5025_singleton__Un__iff,axiom,
    ! [X4: real,A3: set_real,B4: set_real] :
      ( ( ( insert_real @ X4 @ bot_bot_set_real )
        = ( sup_sup_set_real @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bot_set_real )
          & ( B4
            = ( insert_real @ X4 @ bot_bot_set_real ) ) )
        | ( ( A3
            = ( insert_real @ X4 @ bot_bot_set_real ) )
          & ( B4 = bot_bot_set_real ) )
        | ( ( A3
            = ( insert_real @ X4 @ bot_bot_set_real ) )
          & ( B4
            = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5026_singleton__Un__iff,axiom,
    ! [X4: $o,A3: set_o,B4: set_o] :
      ( ( ( insert_o @ X4 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B4
            = ( insert_o @ X4 @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X4 @ bot_bot_set_o ) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X4 @ bot_bot_set_o ) )
          & ( B4
            = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5027_singleton__Un__iff,axiom,
    ! [X4: nat,A3: set_nat,B4: set_nat] :
      ( ( ( insert_nat @ X4 @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X4 @ bot_bot_set_nat ) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X4 @ bot_bot_set_nat ) )
          & ( B4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5028_singleton__Un__iff,axiom,
    ! [X4: int,A3: set_int,B4: set_int] :
      ( ( ( insert_int @ X4 @ bot_bot_set_int )
        = ( sup_sup_set_int @ A3 @ B4 ) )
      = ( ( ( A3 = bot_bot_set_int )
          & ( B4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) )
        | ( ( A3
            = ( insert_int @ X4 @ bot_bot_set_int ) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A3
            = ( insert_int @ X4 @ bot_bot_set_int ) )
          & ( B4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5029_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_eq_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_5030_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_eq_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ M ) @ ( set_or1222409239386451017an_num @ M @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_5031_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_eq_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_5032_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_eq_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_5033_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M )
     => ( ( ord_le3102999989581377725nteger @ M @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_5034_Diff__subset__conv,axiom,
    ! [A3: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B4 ) @ C2 )
      = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_5035_Diff__subset__conv,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B4 ) @ C2 )
      = ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ B4 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_5036_Diff__partition,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B4 @ A3 ) )
        = B4 ) ) ).

% Diff_partition
thf(fact_5037_Diff__partition,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( sup_sup_set_int @ A3 @ ( minus_minus_set_int @ B4 @ A3 ) )
        = B4 ) ) ).

% Diff_partition
thf(fact_5038_in__graphD,axiom,
    ! [K: int,V: int,M: int > option_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ M ) )
     => ( ( M @ K )
        = ( some_int @ V ) ) ) ).

% in_graphD
thf(fact_5039_in__graphD,axiom,
    ! [K: code_integer,V: code_integer,M: code_integer > option_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ K @ V ) @ ( graph_5282091004195177018nteger @ M ) )
     => ( ( M @ K )
        = ( some_Code_integer @ V ) ) ) ).

% in_graphD
thf(fact_5040_in__graphD,axiom,
    ! [K: nat,V: nat,M: nat > option_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% in_graphD
thf(fact_5041_in__graphD,axiom,
    ! [K: vEBT_VEBT,V: nat,M: vEBT_VEBT > option_nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ K @ V ) @ ( graph_VEBT_VEBT_nat @ M ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% in_graphD
thf(fact_5042_in__graphD,axiom,
    ! [K: nat,V: num,M: nat > option_num] :
      ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ K @ V ) @ ( graph_nat_num @ M ) )
     => ( ( M @ K )
        = ( some_num @ V ) ) ) ).

% in_graphD
thf(fact_5043_in__graphI,axiom,
    ! [M: int > option_int,K: int,V: int] :
      ( ( ( M @ K )
        = ( some_int @ V ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ M ) ) ) ).

% in_graphI
thf(fact_5044_in__graphI,axiom,
    ! [M: code_integer > option_Code_integer,K: code_integer,V: code_integer] :
      ( ( ( M @ K )
        = ( some_Code_integer @ V ) )
     => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ K @ V ) @ ( graph_5282091004195177018nteger @ M ) ) ) ).

% in_graphI
thf(fact_5045_in__graphI,axiom,
    ! [M: nat > option_nat,K: nat,V: nat] :
      ( ( ( M @ K )
        = ( some_nat @ V ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) ) ) ).

% in_graphI
thf(fact_5046_in__graphI,axiom,
    ! [M: vEBT_VEBT > option_nat,K: vEBT_VEBT,V: nat] :
      ( ( ( M @ K )
        = ( some_nat @ V ) )
     => ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ K @ V ) @ ( graph_VEBT_VEBT_nat @ M ) ) ) ).

% in_graphI
thf(fact_5047_in__graphI,axiom,
    ! [M: nat > option_num,K: nat,V: num] :
      ( ( ( M @ K )
        = ( some_num @ V ) )
     => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ K @ V ) @ ( graph_nat_num @ M ) ) ) ).

% in_graphI
thf(fact_5048_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_5049_sup__bot__left,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X4 )
      = X4 ) ).

% sup_bot_left
thf(fact_5050_sup__bot__left,axiom,
    ! [X4: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ X4 )
      = X4 ) ).

% sup_bot_left
thf(fact_5051_sup__bot__left,axiom,
    ! [X4: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X4 )
      = X4 ) ).

% sup_bot_left
thf(fact_5052_sup__bot__left,axiom,
    ! [X4: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
      = X4 ) ).

% sup_bot_left
thf(fact_5053_sup__bot__left,axiom,
    ! [X4: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X4 )
      = X4 ) ).

% sup_bot_left
thf(fact_5054_sup__bot__right,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ X4 @ bot_bot_assn )
      = X4 ) ).

% sup_bot_right
thf(fact_5055_sup__bot__right,axiom,
    ! [X4: set_real] :
      ( ( sup_sup_set_real @ X4 @ bot_bot_set_real )
      = X4 ) ).

% sup_bot_right
thf(fact_5056_sup__bot__right,axiom,
    ! [X4: set_o] :
      ( ( sup_sup_set_o @ X4 @ bot_bot_set_o )
      = X4 ) ).

% sup_bot_right
thf(fact_5057_sup__bot__right,axiom,
    ! [X4: set_nat] :
      ( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
      = X4 ) ).

% sup_bot_right
thf(fact_5058_sup__bot__right,axiom,
    ! [X4: set_int] :
      ( ( sup_sup_set_int @ X4 @ bot_bot_set_int )
      = X4 ) ).

% sup_bot_right
thf(fact_5059_bot__eq__sup__iff,axiom,
    ! [X4: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X4 @ Y ) )
      = ( ( X4 = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_5060_bot__eq__sup__iff,axiom,
    ! [X4: set_real,Y: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ X4 @ Y ) )
      = ( ( X4 = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
thf(fact_5061_bot__eq__sup__iff,axiom,
    ! [X4: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X4 @ Y ) )
      = ( ( X4 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_5062_bot__eq__sup__iff,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X4 @ Y ) )
      = ( ( X4 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_5063_bot__eq__sup__iff,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X4 @ Y ) )
      = ( ( X4 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_5064_sup__eq__bot__iff,axiom,
    ! [X4: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X4 @ Y )
        = bot_bot_assn )
      = ( ( X4 = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_5065_sup__eq__bot__iff,axiom,
    ! [X4: set_real,Y: set_real] :
      ( ( ( sup_sup_set_real @ X4 @ Y )
        = bot_bot_set_real )
      = ( ( X4 = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
thf(fact_5066_sup__eq__bot__iff,axiom,
    ! [X4: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X4 @ Y )
        = bot_bot_set_o )
      = ( ( X4 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_5067_sup__eq__bot__iff,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X4 @ Y )
        = bot_bot_set_nat )
      = ( ( X4 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_5068_sup__eq__bot__iff,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X4 @ Y )
        = bot_bot_set_int )
      = ( ( X4 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_5069_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( ( sup_sup_assn @ A @ B )
        = bot_bot_assn )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_5070_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ( sup_sup_set_real @ A @ B )
        = bot_bot_set_real )
      = ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_5071_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_5072_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_5073_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_5074_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_5075_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_5076_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_5077_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_5078_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_5079_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A @ B ) )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_5080_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ A @ B ) )
      = ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_5081_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_5082_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_5083_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_5084_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_5085_sup_Obounded__iff,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A )
      = ( ( ord_less_eq_assn @ B @ A )
        & ( ord_less_eq_assn @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5086_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5087_sup_Obounded__iff,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A )
      = ( ( ord_less_eq_set_int @ B @ A )
        & ( ord_less_eq_set_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5088_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5089_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5090_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_5091_le__sup__iff,axiom,
    ! [X4: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X4 @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5092_le__sup__iff,axiom,
    ! [X4: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X4 @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5093_le__sup__iff,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_set_int @ X4 @ Z )
        & ( ord_less_eq_set_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5094_le__sup__iff,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_rat @ X4 @ Z )
        & ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5095_le__sup__iff,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X4 @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5096_le__sup__iff,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X4 @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X4 @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_5097_sup__bot_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ A @ bot_bot_assn )
      = A ) ).

% sup_bot.right_neutral
thf(fact_5098_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ A @ bot_bot_set_real )
      = A ) ).

% sup_bot.right_neutral
thf(fact_5099_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_5100_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_5101_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_5102_sup__None__1,axiom,
    ! [Y: option_nat] :
      ( ( sup_sup_option_nat @ none_nat @ Y )
      = Y ) ).

% sup_None_1
thf(fact_5103_sup__None__2,axiom,
    ! [X4: option_nat] :
      ( ( sup_sup_option_nat @ X4 @ none_nat )
      = X4 ) ).

% sup_None_2
thf(fact_5104_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S3 ) )
      = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( sup_su6327502436637775413at_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_5105_sup__Un__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ( sup_su2199749269212604124_nat_o
        @ ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ R )
        @ ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ S3 ) )
      = ( ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ ( sup_su6061789376821058069BT_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_5106_sup__Un__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( sup_sup_int_int_o
        @ ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R )
        @ ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ S3 ) )
      = ( ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ ( sup_su6024340866399070445nt_int @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_5107_sup__Un__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ( sup_su2386495712988733984eger_o
        @ ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ R )
        @ ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ S3 ) )
      = ( ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ ( sup_su3575575879904067535nteger @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_5108_sup__Un__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( sup_sup_nat_num_o
        @ ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ R )
        @ ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ S3 ) )
      = ( ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ ( sup_su2042722026077122175at_num @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_5109_star__or__dist1,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A3 @ B4 ) @ C2 )
      = ( sup_sup_assn @ ( times_times_assn @ A3 @ C2 ) @ ( times_times_assn @ B4 @ C2 ) ) ) ).

% star_or_dist1
thf(fact_5110_star__or__dist2,axiom,
    ! [C2: assn,A3: assn,B4: assn] :
      ( ( times_times_assn @ C2 @ ( sup_sup_assn @ A3 @ B4 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C2 @ A3 ) @ ( times_times_assn @ C2 @ B4 ) ) ) ).

% star_or_dist2
thf(fact_5111_ent__disjE,axiom,
    ! [A3: assn,C2: assn,B4: assn] :
      ( ( entails @ A3 @ C2 )
     => ( ( entails @ B4 @ C2 )
       => ( entails @ ( sup_sup_assn @ A3 @ B4 ) @ C2 ) ) ) ).

% ent_disjE
thf(fact_5112_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ P @ R ) ) ).

% ent_disjI1
thf(fact_5113_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ Q @ R ) ) ).

% ent_disjI2
thf(fact_5114_ent__disjI1_H,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( entails @ A3 @ B4 )
     => ( entails @ A3 @ ( sup_sup_assn @ B4 @ C2 ) ) ) ).

% ent_disjI1'
thf(fact_5115_ent__disjI2_H,axiom,
    ! [A3: assn,C2: assn,B4: assn] :
      ( ( entails @ A3 @ C2 )
     => ( entails @ A3 @ ( sup_sup_assn @ B4 @ C2 ) ) ) ).

% ent_disjI2'
thf(fact_5116_ent__disjI1__direct,axiom,
    ! [A3: assn,B4: assn] : ( entails @ A3 @ ( sup_sup_assn @ A3 @ B4 ) ) ).

% ent_disjI1_direct
thf(fact_5117_ent__disjI2__direct,axiom,
    ! [B4: assn,A3: assn] : ( entails @ B4 @ ( sup_sup_assn @ A3 @ B4 ) ) ).

% ent_disjI2_direct
thf(fact_5118_split__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,R: option_nat > assn,Q: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ R )
     => ( ( hoare_7629718768684598413on_nat @ Q @ C @ R )
       => ( hoare_7629718768684598413on_nat @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_5119_split__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,R: $o > assn,Q: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ R )
     => ( ( hoare_hoare_triple_o @ Q @ C @ R )
       => ( hoare_hoare_triple_o @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_5120_split__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,R: vEBT_VEBTi > assn,Q: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ R )
     => ( ( hoare_1429296392585015714_VEBTi @ Q @ C @ R )
       => ( hoare_1429296392585015714_VEBTi @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_5121_split__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,R: nat > assn,Q: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ R )
     => ( ( hoare_3067605981109127869le_nat @ Q @ C @ R )
       => ( hoare_3067605981109127869le_nat @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_5122_norm__assertion__simps_I6_J,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ X4 @ bot_bot_assn )
      = X4 ) ).

% norm_assertion_simps(6)
thf(fact_5123_norm__assertion__simps_I5_J,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X4 )
      = X4 ) ).

% norm_assertion_simps(5)
thf(fact_5124_sup__Un__eq,axiom,
    ! [R: set_VEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( sup_sup_VEBT_VEBT_o
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ R )
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ S3 ) )
      = ( ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( sup_su6272177626956685416T_VEBT @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_5125_sup__Un__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( sup_sup_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S3 ) )
      = ( ^ [X: real] : ( member_real @ X @ ( sup_sup_set_real @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_5126_sup__Un__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( sup_sup_int_o
        @ ^ [X: int] : ( member_int @ X @ R )
        @ ^ [X: int] : ( member_int @ X @ S3 ) )
      = ( ^ [X: int] : ( member_int @ X @ ( sup_sup_set_int @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_5127_sup__Un__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ( sup_sup_set_nat_o
        @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
        @ ^ [X: set_nat] : ( member_set_nat @ X @ S3 ) )
      = ( ^ [X: set_nat] : ( member_set_nat @ X @ ( sup_sup_set_set_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_5128_sup__Un__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_5129_sup__set__def,axiom,
    ( sup_su6272177626956685416T_VEBT
    = ( ^ [A5: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( sup_sup_VEBT_VEBT_o
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A5 )
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5130_sup__set__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( sup_sup_real_o
            @ ^ [X: real] : ( member_real @ X @ A5 )
            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5131_sup__set__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( sup_sup_complex_o
            @ ^ [X: complex] : ( member_complex @ X @ A5 )
            @ ^ [X: complex] : ( member_complex @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5132_sup__set__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( sup_sup_list_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5133_sup__set__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( sup_sup_set_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5134_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X: int] : ( member_int @ X @ A5 )
            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5135_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 )
            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_5136_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_5137_if__rule__split,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q1: option_nat > assn,G: heap_T2636463487746394924on_nat,Q22: option_nat > assn,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q1 ) )
     => ( ( ~ B
         => ( hoare_7629718768684598413on_nat @ P @ G @ Q22 ) )
       => ( ! [X3: option_nat] : ( entails @ ( sup_sup_assn @ ( times_times_assn @ ( Q1 @ X3 ) @ ( pure_assn @ B ) ) @ ( times_times_assn @ ( Q22 @ X3 ) @ ( pure_assn @ ~ B ) ) ) @ ( Q @ X3 ) )
         => ( hoare_7629718768684598413on_nat @ P @ ( if_Hea5867803462524415986on_nat @ B @ F @ G ) @ Q ) ) ) ) ).

% if_rule_split
thf(fact_5138_if__rule__split,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q1: $o > assn,G: heap_Time_Heap_o,Q22: $o > assn,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q1 ) )
     => ( ( ~ B
         => ( hoare_hoare_triple_o @ P @ G @ Q22 ) )
       => ( ! [X3: $o] : ( entails @ ( sup_sup_assn @ ( times_times_assn @ ( Q1 @ X3 ) @ ( pure_assn @ B ) ) @ ( times_times_assn @ ( Q22 @ X3 ) @ ( pure_assn @ ~ B ) ) ) @ ( Q @ X3 ) )
         => ( hoare_hoare_triple_o @ P @ ( if_Heap_Time_Heap_o @ B @ F @ G ) @ Q ) ) ) ) ).

% if_rule_split
thf(fact_5139_if__rule__split,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q1: vEBT_VEBTi > assn,G: heap_T8145700208782473153_VEBTi,Q22: vEBT_VEBTi > assn,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q1 ) )
     => ( ( ~ B
         => ( hoare_1429296392585015714_VEBTi @ P @ G @ Q22 ) )
       => ( ! [X3: vEBT_VEBTi] : ( entails @ ( sup_sup_assn @ ( times_times_assn @ ( Q1 @ X3 ) @ ( pure_assn @ B ) ) @ ( times_times_assn @ ( Q22 @ X3 ) @ ( pure_assn @ ~ B ) ) ) @ ( Q @ X3 ) )
         => ( hoare_1429296392585015714_VEBTi @ P @ ( if_Hea8453224502484754311_VEBTi @ B @ F @ G ) @ Q ) ) ) ) ).

% if_rule_split
thf(fact_5140_if__rule__split,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q1: nat > assn,G: heap_Time_Heap_nat,Q22: nat > assn,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q1 ) )
     => ( ( ~ B
         => ( hoare_3067605981109127869le_nat @ P @ G @ Q22 ) )
       => ( ! [X3: nat] : ( entails @ ( sup_sup_assn @ ( times_times_assn @ ( Q1 @ X3 ) @ ( pure_assn @ B ) ) @ ( times_times_assn @ ( Q22 @ X3 ) @ ( pure_assn @ ~ B ) ) ) @ ( Q @ X3 ) )
         => ( hoare_3067605981109127869le_nat @ P @ ( if_Hea2662716070787841314ap_nat @ B @ F @ G ) @ Q ) ) ) ) ).

% if_rule_split
thf(fact_5141_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X4: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5142_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X4: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5143_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X4: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5144_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X4: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5145_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5146_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X4: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X4 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_5147_inf__sup__ord_I3_J,axiom,
    ! [X4: assn,Y: assn] : ( ord_less_eq_assn @ X4 @ ( sup_sup_assn @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5148_inf__sup__ord_I3_J,axiom,
    ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5149_inf__sup__ord_I3_J,axiom,
    ! [X4: set_int,Y: set_int] : ( ord_less_eq_set_int @ X4 @ ( sup_sup_set_int @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5150_inf__sup__ord_I3_J,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ X4 @ ( sup_sup_rat @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5151_inf__sup__ord_I3_J,axiom,
    ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5152_inf__sup__ord_I3_J,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ X4 @ ( sup_sup_int @ X4 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_5153_le__supE,axiom,
    ! [A: assn,B: assn,X4: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_assn @ A @ X4 )
         => ~ ( ord_less_eq_assn @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5154_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_set_nat @ A @ X4 )
         => ~ ( ord_less_eq_set_nat @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5155_le__supE,axiom,
    ! [A: set_int,B: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_set_int @ A @ X4 )
         => ~ ( ord_less_eq_set_int @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5156_le__supE,axiom,
    ! [A: rat,B: rat,X4: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_rat @ A @ X4 )
         => ~ ( ord_less_eq_rat @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5157_le__supE,axiom,
    ! [A: nat,B: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_nat @ A @ X4 )
         => ~ ( ord_less_eq_nat @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5158_le__supE,axiom,
    ! [A: int,B: int,X4: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X4 )
     => ~ ( ( ord_less_eq_int @ A @ X4 )
         => ~ ( ord_less_eq_int @ B @ X4 ) ) ) ).

% le_supE
thf(fact_5159_le__supI,axiom,
    ! [A: assn,X4: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X4 )
     => ( ( ord_less_eq_assn @ B @ X4 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5160_le__supI,axiom,
    ! [A: set_nat,X4: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X4 )
     => ( ( ord_less_eq_set_nat @ B @ X4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5161_le__supI,axiom,
    ! [A: set_int,X4: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ X4 )
     => ( ( ord_less_eq_set_int @ B @ X4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5162_le__supI,axiom,
    ! [A: rat,X4: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X4 )
     => ( ( ord_less_eq_rat @ B @ X4 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5163_le__supI,axiom,
    ! [A: nat,X4: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X4 )
     => ( ( ord_less_eq_nat @ B @ X4 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5164_le__supI,axiom,
    ! [A: int,X4: int,B: int] :
      ( ( ord_less_eq_int @ A @ X4 )
     => ( ( ord_less_eq_int @ B @ X4 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X4 ) ) ) ).

% le_supI
thf(fact_5165_sup__ge1,axiom,
    ! [X4: assn,Y: assn] : ( ord_less_eq_assn @ X4 @ ( sup_sup_assn @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5166_sup__ge1,axiom,
    ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5167_sup__ge1,axiom,
    ! [X4: set_int,Y: set_int] : ( ord_less_eq_set_int @ X4 @ ( sup_sup_set_int @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5168_sup__ge1,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ X4 @ ( sup_sup_rat @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5169_sup__ge1,axiom,
    ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5170_sup__ge1,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ X4 @ ( sup_sup_int @ X4 @ Y ) ) ).

% sup_ge1
thf(fact_5171_sup__ge2,axiom,
    ! [Y: assn,X4: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5172_sup__ge2,axiom,
    ! [Y: set_nat,X4: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5173_sup__ge2,axiom,
    ! [Y: set_int,X4: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5174_sup__ge2,axiom,
    ! [Y: rat,X4: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5175_sup__ge2,axiom,
    ! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5176_sup__ge2,axiom,
    ! [Y: int,X4: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X4 @ Y ) ) ).

% sup_ge2
thf(fact_5177_le__supI1,axiom,
    ! [X4: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X4 @ A )
     => ( ord_less_eq_assn @ X4 @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI1
thf(fact_5178_le__supI1,axiom,
    ! [X4: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ A )
     => ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_5179_le__supI1,axiom,
    ! [X4: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ A )
     => ( ord_less_eq_set_int @ X4 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_5180_le__supI1,axiom,
    ! [X4: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X4 @ A )
     => ( ord_less_eq_rat @ X4 @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI1
thf(fact_5181_le__supI1,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X4 @ A )
     => ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_5182_le__supI1,axiom,
    ! [X4: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X4 @ A )
     => ( ord_less_eq_int @ X4 @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_5183_le__supI2,axiom,
    ! [X4: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ X4 @ B )
     => ( ord_less_eq_assn @ X4 @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI2
thf(fact_5184_le__supI2,axiom,
    ! [X4: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ B )
     => ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_5185_le__supI2,axiom,
    ! [X4: set_int,B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ B )
     => ( ord_less_eq_set_int @ X4 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_5186_le__supI2,axiom,
    ! [X4: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ X4 @ B )
     => ( ord_less_eq_rat @ X4 @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI2
thf(fact_5187_le__supI2,axiom,
    ! [X4: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X4 @ B )
     => ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_5188_le__supI2,axiom,
    ! [X4: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X4 @ B )
     => ( ord_less_eq_int @ X4 @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_5189_sup_Omono,axiom,
    ! [C: assn,A: assn,D: assn,B: assn] :
      ( ( ord_less_eq_assn @ C @ A )
     => ( ( ord_less_eq_assn @ D @ B )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C @ D ) @ ( sup_sup_assn @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5190_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ D @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5191_sup_Omono,axiom,
    ! [C: set_int,A: set_int,D: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C @ A )
     => ( ( ord_less_eq_set_int @ D @ B )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D ) @ ( sup_sup_set_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5192_sup_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5193_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5194_sup_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_5195_sup__mono,axiom,
    ! [A: assn,C: assn,B: assn,D: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ( ord_less_eq_assn @ B @ D )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ ( sup_sup_assn @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5196_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5197_sup__mono,axiom,
    ! [A: set_int,C: set_int,B: set_int,D: set_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ( ord_less_eq_set_int @ B @ D )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ ( sup_sup_set_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5198_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5199_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5200_sup__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_5201_sup__least,axiom,
    ! [Y: assn,X4: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X4 )
     => ( ( ord_less_eq_assn @ Z @ X4 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5202_sup__least,axiom,
    ! [Y: set_nat,X4: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X4 )
     => ( ( ord_less_eq_set_nat @ Z @ X4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5203_sup__least,axiom,
    ! [Y: set_int,X4: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ( ( ord_less_eq_set_int @ Z @ X4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5204_sup__least,axiom,
    ! [Y: rat,X4: rat,Z: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ( ord_less_eq_rat @ Z @ X4 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5205_sup__least,axiom,
    ! [Y: nat,X4: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ( ( ord_less_eq_nat @ Z @ X4 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5206_sup__least,axiom,
    ! [Y: int,X4: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ( ( ord_less_eq_int @ Z @ X4 )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X4 ) ) ) ).

% sup_least
thf(fact_5207_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X: assn,Y4: assn] :
          ( ( sup_sup_assn @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5208_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y4: set_nat] :
          ( ( sup_sup_set_nat @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5209_le__iff__sup,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y4: set_int] :
          ( ( sup_sup_set_int @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5210_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y4: rat] :
          ( ( sup_sup_rat @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5211_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( sup_sup_nat @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5212_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y4: int] :
          ( ( sup_sup_int @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_5213_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( A
        = ( sup_sup_assn @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5214_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5215_sup_OorderE,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( A
        = ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5216_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5217_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5218_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_5219_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( sup_sup_assn @ A @ B ) )
     => ( ord_less_eq_assn @ B @ A ) ) ).

% sup.orderI
thf(fact_5220_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_5221_sup_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A
        = ( sup_sup_set_int @ A @ B ) )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% sup.orderI
thf(fact_5222_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_5223_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_5224_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_5225_sup__unique,axiom,
    ! [F: assn > assn > assn,X4: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: assn,Y3: assn,Z3: assn] :
              ( ( ord_less_eq_assn @ Y3 @ X3 )
             => ( ( ord_less_eq_assn @ Z3 @ X3 )
               => ( ord_less_eq_assn @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_assn @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5226_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X4: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_nat,Y3: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z3 @ X3 )
               => ( ord_less_eq_set_nat @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5227_sup__unique,axiom,
    ! [F: set_int > set_int > set_int,X4: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_int,Y3: set_int,Z3: set_int] :
              ( ( ord_less_eq_set_int @ Y3 @ X3 )
             => ( ( ord_less_eq_set_int @ Z3 @ X3 )
               => ( ord_less_eq_set_int @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_int @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5228_sup__unique,axiom,
    ! [F: rat > rat > rat,X4: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: rat,Y3: rat,Z3: rat] :
              ( ( ord_less_eq_rat @ Y3 @ X3 )
             => ( ( ord_less_eq_rat @ Z3 @ X3 )
               => ( ord_less_eq_rat @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_rat @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5229_sup__unique,axiom,
    ! [F: nat > nat > nat,X4: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z3 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5230_sup__unique,axiom,
    ! [F: int > int > int,X4: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z3: int] :
              ( ( ord_less_eq_int @ Y3 @ X3 )
             => ( ( ord_less_eq_int @ Z3 @ X3 )
               => ( ord_less_eq_int @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_int @ X4 @ Y )
            = ( F @ X4 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_5231_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5232_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5233_sup_Oabsorb1,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( sup_sup_set_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5234_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5235_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5236_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_5237_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5238_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5239_sup_Oabsorb2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( sup_sup_set_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5240_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5241_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5242_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_5243_sup__absorb1,axiom,
    ! [Y: assn,X4: assn] :
      ( ( ord_less_eq_assn @ Y @ X4 )
     => ( ( sup_sup_assn @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5244_sup__absorb1,axiom,
    ! [Y: set_nat,X4: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X4 )
     => ( ( sup_sup_set_nat @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5245_sup__absorb1,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ( ( sup_sup_set_int @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5246_sup__absorb1,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ( sup_sup_rat @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5247_sup__absorb1,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ( ( sup_sup_nat @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5248_sup__absorb1,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ( ( sup_sup_int @ X4 @ Y )
        = X4 ) ) ).

% sup_absorb1
thf(fact_5249_sup__absorb2,axiom,
    ! [X4: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X4 @ Y )
     => ( ( sup_sup_assn @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5250_sup__absorb2,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X4 @ Y )
     => ( ( sup_sup_set_nat @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5251_sup__absorb2,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( sup_sup_set_int @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5252_sup__absorb2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( sup_sup_rat @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5253_sup__absorb2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( sup_sup_nat @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5254_sup__absorb2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( sup_sup_int @ X4 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_5255_sup_OboundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_assn @ B @ A )
         => ~ ( ord_less_eq_assn @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5256_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5257_sup_OboundedE,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_int @ B @ A )
         => ~ ( ord_less_eq_set_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5258_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5259_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5260_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_5261_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C @ A )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5262_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5263_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ A )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5264_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5265_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5266_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_5267_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( A2
          = ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5268_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( A2
          = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5269_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B2: set_int,A2: set_int] :
          ( A2
          = ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5270_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( sup_sup_rat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5271_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5272_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_5273_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ A @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5274_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5275_sup_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : ( ord_less_eq_set_int @ A @ ( sup_sup_set_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5276_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5277_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5278_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_5279_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : ( ord_less_eq_assn @ B @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5280_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5281_sup_Ocobounded2,axiom,
    ! [B: set_int,A: set_int] : ( ord_less_eq_set_int @ B @ ( sup_sup_set_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5282_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5283_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5284_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_5285_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5286_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5287_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B2: set_int,A2: set_int] :
          ( ( sup_sup_set_int @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5288_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( sup_sup_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5289_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5290_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_5291_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5292_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5293_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A2: set_int,B2: set_int] :
          ( ( sup_sup_set_int @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5294_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( sup_sup_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5295_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5296_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_5297_sup_OcoboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ C @ A )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5298_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5299_sup_OcoboundedI1,axiom,
    ! [C: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C @ A )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5300_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5301_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5302_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_5303_sup_OcoboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ C @ B )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5304_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5305_sup_OcoboundedI2,axiom,
    ! [C: set_int,B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ C @ B )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5306_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5307_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5308_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_5309_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ C @ B )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5310_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C @ B )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5311_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5312_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5313_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5314_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5315_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ C @ A )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5316_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C @ A )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5317_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5318_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5319_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5320_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5321_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( A2
            = ( sup_sup_assn @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5322_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( A2
            = ( sup_sup_set_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5323_sup_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( A2
            = ( sup_sup_real @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5324_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( sup_sup_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5325_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( sup_sup_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5326_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( sup_sup_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5327_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B @ C ) @ A )
     => ~ ( ( ord_less_assn @ B @ A )
         => ~ ( ord_less_assn @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5328_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5329_sup_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( sup_sup_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5330_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5331_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5332_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5333_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5334_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5335_sup_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( sup_sup_real @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5336_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5337_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5338_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5339_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5340_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5341_sup_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( sup_sup_real @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5342_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5343_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5344_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5345_less__supI2,axiom,
    ! [X4: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ X4 @ B )
     => ( ord_less_assn @ X4 @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI2
thf(fact_5346_less__supI2,axiom,
    ! [X4: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X4 @ B )
     => ( ord_less_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5347_less__supI2,axiom,
    ! [X4: real,B: real,A: real] :
      ( ( ord_less_real @ X4 @ B )
     => ( ord_less_real @ X4 @ ( sup_sup_real @ A @ B ) ) ) ).

% less_supI2
thf(fact_5348_less__supI2,axiom,
    ! [X4: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ X4 @ B )
     => ( ord_less_rat @ X4 @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5349_less__supI2,axiom,
    ! [X4: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X4 @ B )
     => ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5350_less__supI2,axiom,
    ! [X4: int,B: int,A: int] :
      ( ( ord_less_int @ X4 @ B )
     => ( ord_less_int @ X4 @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI2
thf(fact_5351_less__supI1,axiom,
    ! [X4: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ X4 @ A )
     => ( ord_less_assn @ X4 @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI1
thf(fact_5352_less__supI1,axiom,
    ! [X4: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ X4 @ A )
     => ( ord_less_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5353_less__supI1,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( ord_less_real @ X4 @ A )
     => ( ord_less_real @ X4 @ ( sup_sup_real @ A @ B ) ) ) ).

% less_supI1
thf(fact_5354_less__supI1,axiom,
    ! [X4: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ X4 @ A )
     => ( ord_less_rat @ X4 @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5355_less__supI1,axiom,
    ! [X4: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X4 @ A )
     => ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5356_less__supI1,axiom,
    ! [X4: int,A: int,B: int] :
      ( ( ord_less_int @ X4 @ A )
     => ( ord_less_int @ X4 @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI1
thf(fact_5357_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_5358_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_5359_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q5: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q5 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q5 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5360_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila8603258263270017530r_assn @ sup_sup_assn @ bot_bot_assn
    @ ^ [X: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ X )
    @ ^ [X: assn,Y4: assn] : ( ord_less_assn @ Y4 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_5361_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila4459961161675019956t_real @ sup_sup_set_real @ bot_bot_set_real
    @ ^ [X: set_real,Y4: set_real] : ( ord_less_eq_set_real @ Y4 @ X )
    @ ^ [X: set_real,Y4: set_real] : ( ord_less_set_real @ Y4 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_5362_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila2554085542299052326_set_o @ sup_sup_set_o @ bot_bot_set_o
    @ ^ [X: set_o,Y4: set_o] : ( ord_less_eq_set_o @ Y4 @ X )
    @ ^ [X: set_o,Y4: set_o] : ( ord_less_set_o @ Y4 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_5363_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila1667268886620078168et_nat @ sup_sup_set_nat @ bot_bot_set_nat
    @ ^ [X: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X )
    @ ^ [X: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_5364_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila6712789903965657268et_int @ sup_sup_set_int @ bot_bot_set_int
    @ ^ [X: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ X )
    @ ^ [X: set_int,Y4: set_int] : ( ord_less_set_int @ Y4 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_5365_merge__correct,axiom,
    ! [L1: list_real,L22: list_real] :
      ( ( ( distinct_real @ L1 )
        & ( sorted_wrt_real @ ord_less_eq_real @ L1 ) )
     => ( ( ( distinct_real @ L22 )
          & ( sorted_wrt_real @ ord_less_eq_real @ L22 ) )
       => ( ( distinct_real @ ( merge_real @ L1 @ L22 ) )
          & ( sorted_wrt_real @ ord_less_eq_real @ ( merge_real @ L1 @ L22 ) )
          & ( ( set_real2 @ ( merge_real @ L1 @ L22 ) )
            = ( sup_sup_set_real @ ( set_real2 @ L1 ) @ ( set_real2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5366_merge__correct,axiom,
    ! [L1: list_o,L22: list_o] :
      ( ( ( distinct_o @ L1 )
        & ( sorted_wrt_o @ ord_less_eq_o @ L1 ) )
     => ( ( ( distinct_o @ L22 )
          & ( sorted_wrt_o @ ord_less_eq_o @ L22 ) )
       => ( ( distinct_o @ ( merge_o @ L1 @ L22 ) )
          & ( sorted_wrt_o @ ord_less_eq_o @ ( merge_o @ L1 @ L22 ) )
          & ( ( set_o2 @ ( merge_o @ L1 @ L22 ) )
            = ( sup_sup_set_o @ ( set_o2 @ L1 ) @ ( set_o2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5367_merge__correct,axiom,
    ! [L1: list_rat,L22: list_rat] :
      ( ( ( distinct_rat @ L1 )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ L1 ) )
     => ( ( ( distinct_rat @ L22 )
          & ( sorted_wrt_rat @ ord_less_eq_rat @ L22 ) )
       => ( ( distinct_rat @ ( merge_rat @ L1 @ L22 ) )
          & ( sorted_wrt_rat @ ord_less_eq_rat @ ( merge_rat @ L1 @ L22 ) )
          & ( ( set_rat2 @ ( merge_rat @ L1 @ L22 ) )
            = ( sup_sup_set_rat @ ( set_rat2 @ L1 ) @ ( set_rat2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5368_merge__correct,axiom,
    ! [L1: list_num,L22: list_num] :
      ( ( ( distinct_num @ L1 )
        & ( sorted_wrt_num @ ord_less_eq_num @ L1 ) )
     => ( ( ( distinct_num @ L22 )
          & ( sorted_wrt_num @ ord_less_eq_num @ L22 ) )
       => ( ( distinct_num @ ( merge_num @ L1 @ L22 ) )
          & ( sorted_wrt_num @ ord_less_eq_num @ ( merge_num @ L1 @ L22 ) )
          & ( ( set_num2 @ ( merge_num @ L1 @ L22 ) )
            = ( sup_sup_set_num @ ( set_num2 @ L1 ) @ ( set_num2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5369_merge__correct,axiom,
    ! [L1: list_nat,L22: list_nat] :
      ( ( ( distinct_nat @ L1 )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ L1 ) )
     => ( ( ( distinct_nat @ L22 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ L22 ) )
       => ( ( distinct_nat @ ( merge_nat @ L1 @ L22 ) )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ ( merge_nat @ L1 @ L22 ) )
          & ( ( set_nat2 @ ( merge_nat @ L1 @ L22 ) )
            = ( sup_sup_set_nat @ ( set_nat2 @ L1 ) @ ( set_nat2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5370_merge__correct,axiom,
    ! [L1: list_int,L22: list_int] :
      ( ( ( distinct_int @ L1 )
        & ( sorted_wrt_int @ ord_less_eq_int @ L1 ) )
     => ( ( ( distinct_int @ L22 )
          & ( sorted_wrt_int @ ord_less_eq_int @ L22 ) )
       => ( ( distinct_int @ ( merge_int @ L1 @ L22 ) )
          & ( sorted_wrt_int @ ord_less_eq_int @ ( merge_int @ L1 @ L22 ) )
          & ( ( set_int2 @ ( merge_int @ L1 @ L22 ) )
            = ( sup_sup_set_int @ ( set_int2 @ L1 ) @ ( set_int2 @ L22 ) ) ) ) ) ) ).

% merge_correct
thf(fact_5371_time__array__nth,axiom,
    ! [P5: array_VEBT_VEBTi,I: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( ( time_f8309030937046825446_VEBTi @ ( array_nth_VEBT_VEBTi @ P5 @ I ) @ H2 )
       => ( ( time_time_VEBT_VEBTi @ ( array_nth_VEBT_VEBTi @ P5 @ I ) @ H2 )
          = zero_zero_nat ) )
      & ( ~ ( time_f8309030937046825446_VEBTi @ ( array_nth_VEBT_VEBTi @ P5 @ I ) @ H2 )
       => ( ( time_time_VEBT_VEBTi @ ( array_nth_VEBT_VEBTi @ P5 @ I ) @ H2 )
          = one_one_nat ) ) ) ).

% time_array_nth
thf(fact_5372_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_5373_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_5374_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_5375_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_5376_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_5377_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_5378_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_5379_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_5380_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_5381_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_5382_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_5383_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_5384_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_5385_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_5386_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_5387_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_5388_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_5389_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_5390_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_5391_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_5392_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_5393_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_5394_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_5395_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_5396_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_5397_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_5398_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_5399_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_5400_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_5401_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_5402_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_5403_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_5404_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_5405_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5406_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5407_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5408_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5409_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_5410_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_5411_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_5412_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_5413_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_5414_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_5415_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_5416_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_5417_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_5418_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_5419_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_5420_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_5421_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5422_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5423_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5424_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5425_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_5426_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_5427_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_5428_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_5429_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_5430_norm__assertion__simps_I15_J,axiom,
    ! [X4: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X4 @ Y ) @ Z )
      = ( sup_sup_assn @ X4 @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(15)
thf(fact_5431_norm__assertion__simps_I32_J,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ X4 @ X4 )
      = X4 ) ).

% norm_assertion_simps(32)
thf(fact_5432_assn__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X: assn,Y4: assn] : ( sup_sup_assn @ Y4 @ X ) ) ) ).

% assn_aci(5)
thf(fact_5433_assn__aci_I7_J,axiom,
    ! [X4: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X4 @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X4 @ Z ) ) ) ).

% assn_aci(7)
thf(fact_5434_assn__aci_I8_J,axiom,
    ! [X4: assn,Y: assn] :
      ( ( sup_sup_assn @ X4 @ ( sup_sup_assn @ X4 @ Y ) )
      = ( sup_sup_assn @ X4 @ Y ) ) ).

% assn_aci(8)
thf(fact_5435_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5436_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_5437_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5438_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5439_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5440_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5441_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5442_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5443_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5444_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_5445_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_5446_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_5447_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_5448_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5449_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5450_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5451_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5452_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_5453_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_5454_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5455_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_5456_abs__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_mult
thf(fact_5457_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5458_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5459_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5460_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5461_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_5462_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5463_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_5464_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5465_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5466_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5467_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_5468_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5469_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5470_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5471_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5472_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5473_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5474_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5475_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5476_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5477_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5478_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5479_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5480_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5481_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5482_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5483_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5484_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5485_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5486_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5487_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5488_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5489_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5490_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5491_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5492_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5493_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5494_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M5: int] :
          ? [N4: int] :
            ( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N4 ) )
            & ( member_int @ N4 @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_5495_dense__eq0__I,axiom,
    ! [X4: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ E ) )
     => ( X4 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_5496_dense__eq0__I,axiom,
    ! [X4: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ E ) )
     => ( X4 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5497_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5498_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5499_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5500_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5501_abs__mult__pos,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X4 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5502_abs__mult__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X4 )
        = ( abs_abs_real @ ( times_times_real @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5503_abs__mult__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X4 )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5504_abs__mult__pos,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X4 )
        = ( abs_abs_int @ ( times_times_int @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_5505_merge_Osimps_I3_J,axiom,
    ! [X1: real,X2: real,L1: list_real,L22: list_real] :
      ( ( ( ord_less_real @ X1 @ X2 )
       => ( ( merge_real @ ( cons_real @ X1 @ L1 ) @ ( cons_real @ X2 @ L22 ) )
          = ( cons_real @ X1 @ ( merge_real @ L1 @ ( cons_real @ X2 @ L22 ) ) ) ) )
      & ( ~ ( ord_less_real @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( merge_real @ ( cons_real @ X1 @ L1 ) @ ( cons_real @ X2 @ L22 ) )
              = ( cons_real @ X1 @ ( merge_real @ L1 @ L22 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( merge_real @ ( cons_real @ X1 @ L1 ) @ ( cons_real @ X2 @ L22 ) )
              = ( cons_real @ X2 @ ( merge_real @ ( cons_real @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_5506_merge_Osimps_I3_J,axiom,
    ! [X1: rat,X2: rat,L1: list_rat,L22: list_rat] :
      ( ( ( ord_less_rat @ X1 @ X2 )
       => ( ( merge_rat @ ( cons_rat @ X1 @ L1 ) @ ( cons_rat @ X2 @ L22 ) )
          = ( cons_rat @ X1 @ ( merge_rat @ L1 @ ( cons_rat @ X2 @ L22 ) ) ) ) )
      & ( ~ ( ord_less_rat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( merge_rat @ ( cons_rat @ X1 @ L1 ) @ ( cons_rat @ X2 @ L22 ) )
              = ( cons_rat @ X1 @ ( merge_rat @ L1 @ L22 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( merge_rat @ ( cons_rat @ X1 @ L1 ) @ ( cons_rat @ X2 @ L22 ) )
              = ( cons_rat @ X2 @ ( merge_rat @ ( cons_rat @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_5507_merge_Osimps_I3_J,axiom,
    ! [X1: num,X2: num,L1: list_num,L22: list_num] :
      ( ( ( ord_less_num @ X1 @ X2 )
       => ( ( merge_num @ ( cons_num @ X1 @ L1 ) @ ( cons_num @ X2 @ L22 ) )
          = ( cons_num @ X1 @ ( merge_num @ L1 @ ( cons_num @ X2 @ L22 ) ) ) ) )
      & ( ~ ( ord_less_num @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( merge_num @ ( cons_num @ X1 @ L1 ) @ ( cons_num @ X2 @ L22 ) )
              = ( cons_num @ X1 @ ( merge_num @ L1 @ L22 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( merge_num @ ( cons_num @ X1 @ L1 ) @ ( cons_num @ X2 @ L22 ) )
              = ( cons_num @ X2 @ ( merge_num @ ( cons_num @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_5508_merge_Osimps_I3_J,axiom,
    ! [X1: nat,X2: nat,L1: list_nat,L22: list_nat] :
      ( ( ( ord_less_nat @ X1 @ X2 )
       => ( ( merge_nat @ ( cons_nat @ X1 @ L1 ) @ ( cons_nat @ X2 @ L22 ) )
          = ( cons_nat @ X1 @ ( merge_nat @ L1 @ ( cons_nat @ X2 @ L22 ) ) ) ) )
      & ( ~ ( ord_less_nat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( merge_nat @ ( cons_nat @ X1 @ L1 ) @ ( cons_nat @ X2 @ L22 ) )
              = ( cons_nat @ X1 @ ( merge_nat @ L1 @ L22 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( merge_nat @ ( cons_nat @ X1 @ L1 ) @ ( cons_nat @ X2 @ L22 ) )
              = ( cons_nat @ X2 @ ( merge_nat @ ( cons_nat @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_5509_merge_Osimps_I3_J,axiom,
    ! [X1: int,X2: int,L1: list_int,L22: list_int] :
      ( ( ( ord_less_int @ X1 @ X2 )
       => ( ( merge_int @ ( cons_int @ X1 @ L1 ) @ ( cons_int @ X2 @ L22 ) )
          = ( cons_int @ X1 @ ( merge_int @ L1 @ ( cons_int @ X2 @ L22 ) ) ) ) )
      & ( ~ ( ord_less_int @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( merge_int @ ( cons_int @ X1 @ L1 ) @ ( cons_int @ X2 @ L22 ) )
              = ( cons_int @ X1 @ ( merge_int @ L1 @ L22 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( merge_int @ ( cons_int @ X1 @ L1 ) @ ( cons_int @ X2 @ L22 ) )
              = ( cons_int @ X2 @ ( merge_int @ ( cons_int @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_5510_abs__diff__le__iff,axiom,
    ! [X4: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R3 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X4 )
        & ( ord_le3102999989581377725nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5511_abs__diff__le__iff,axiom,
    ! [X4: real,A: real,R3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R3 ) @ X4 )
        & ( ord_less_eq_real @ X4 @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5512_abs__diff__le__iff,axiom,
    ! [X4: rat,A: rat,R3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R3 ) @ X4 )
        & ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5513_abs__diff__le__iff,axiom,
    ! [X4: int,A: int,R3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R3 ) @ X4 )
        & ( ord_less_eq_int @ X4 @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5514_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5515_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5516_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5517_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5518_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5519_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5520_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5521_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5522_abs__diff__less__iff,axiom,
    ! [X4: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R3 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X4 )
        & ( ord_le6747313008572928689nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5523_abs__diff__less__iff,axiom,
    ! [X4: real,A: real,R3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R3 ) @ X4 )
        & ( ord_less_real @ X4 @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5524_abs__diff__less__iff,axiom,
    ! [X4: rat,A: rat,R3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R3 ) @ X4 )
        & ( ord_less_rat @ X4 @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5525_abs__diff__less__iff,axiom,
    ! [X4: int,A: int,R3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R3 ) @ X4 )
        & ( ord_less_int @ X4 @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5526_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q5: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q5 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5527_abs__add__one__gt__zero,axiom,
    ! [X4: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5528_abs__add__one__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5529_abs__add__one__gt__zero,axiom,
    ! [X4: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5530_abs__add__one__gt__zero,axiom,
    ! [X4: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5531_time__refines,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,C3: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( refine5565527176597971370_VEBTi @ C @ C3 )
     => ( ~ ( time_f8309030937046825446_VEBTi @ C3 @ H2 )
       => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ C @ H2 ) @ ( time_time_VEBT_VEBTi @ C3 @ H2 ) ) ) ) ).

% time_refines
thf(fact_5532_time__refines,axiom,
    ! [C: heap_Time_Heap_o,C3: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit] :
      ( ( refine_Imp_refines_o @ C @ C3 )
     => ( ~ ( time_fails_o @ C3 @ H2 )
       => ( ord_less_eq_nat @ ( time_time_o @ C @ H2 ) @ ( time_time_o @ C3 @ H2 ) ) ) ) ).

% time_refines
thf(fact_5533_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_5534_decr__lemma,axiom,
    ! [D: int,X4: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_5535_incr__lemma,axiom,
    ! [D: int,Z: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_5536_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_5537_bezw__0,axiom,
    ! [X4: nat] :
      ( ( bezw @ X4 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5538_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L3: int,K3: int,Q7: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A33
                = ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
              & ( L3 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q7 @ L3 ) ) )
          | ? [R2: int,L3: int,K3: int,Q7: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A33
                = ( product_Pair_int_int @ Q7 @ R2 ) )
              & ( ( sgn_sgn_int @ R2 )
                = ( sgn_sgn_int @ L3 ) )
              & ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L3 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q7 @ L3 ) @ R2 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5539_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A32 )
     => ( ( ( A22 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q4: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q4 @ A22 ) ) ) )
         => ~ ! [R4: int,Q4: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q4 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q4 @ A22 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5540_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( linord2324613341767563021nteger @ ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) ) )
        = ( remove1_Code_integer @ X4 @ ( linord2324613341767563021nteger @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5541_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( linord4252657396651189596t_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
        = ( remove1_real @ X4 @ ( linord4252657396651189596t_real @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5542_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
        = ( remove1_o @ X4 @ ( linord3142498349692569832_set_o @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5543_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
        = ( remove1_int @ X4 @ ( linord2612477271533052124et_int @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5544_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
        = ( remove1_nat @ X4 @ ( linord2614967742042102400et_nat @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5545_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_5546_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_5547_sgn__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( sgn_sgn_Code_integer @ A ) ) ).

% sgn_sgn
thf(fact_5548_sgn__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% sgn_sgn
thf(fact_5549_sgn__0,axiom,
    ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% sgn_0
thf(fact_5550_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_5551_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_5552_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_5553_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_5554_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_5555_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_5556_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_5557_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_5558_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_5559_sgn__greater,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_greater
thf(fact_5560_sgn__greater,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_greater
thf(fact_5561_sgn__greater,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_greater
thf(fact_5562_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_5563_sgn__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_less
thf(fact_5564_sgn__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_less
thf(fact_5565_sgn__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_less
thf(fact_5566_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_5567_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_5568_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_5569_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_5570_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_5571_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_5572_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_5573_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_5574_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
thf(fact_5575_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_5576_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_5577_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_5578_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
thf(fact_5579_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_5580_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_5581_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_5582_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_5583_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5584_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5585_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5586_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5587_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).

% sgn_mult
thf(fact_5588_sgn__mult,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_mult
thf(fact_5589_sgn__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_mult
thf(fact_5590_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_5591_sgn__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).

% sgn_mult
thf(fact_5592_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5593_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5594_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5595_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5596_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5597_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5598_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5599_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5600_abs__mult__sgn,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5601_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5602_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5603_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5604_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5605_sgn__mult__abs,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5606_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5607_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5608_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5609_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5610_mult__sgn__abs,axiom,
    ! [X4: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_5611_mult__sgn__abs,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( abs_abs_real @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_5612_mult__sgn__abs,axiom,
    ! [X4: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X4 ) @ ( abs_abs_rat @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_5613_mult__sgn__abs,axiom,
    ! [X4: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X4 ) @ ( abs_abs_int @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_5614_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_rat] : ( sorted_wrt_rat @ ord_less_eq_rat @ ( linord1979837681955606664et_rat @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5615_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_num] : ( sorted_wrt_num @ ord_less_eq_num @ ( linord8395671565052656842et_num @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5616_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( linord2612477271533052124et_int @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5617_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5618_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_real] : ( sorted_wrt_real @ ord_less_real @ ( linord4252657396651189596t_real @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5619_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_rat] : ( sorted_wrt_rat @ ord_less_rat @ ( linord1979837681955606664et_rat @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5620_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_num] : ( sorted_wrt_num @ ord_less_num @ ( linord8395671565052656842et_num @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5621_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_int] : ( sorted_wrt_int @ ord_less_int @ ( linord2612477271533052124et_int @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5622_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5623_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_1_pos
thf(fact_5624_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_1_pos
thf(fact_5625_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_1_pos
thf(fact_5626_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_5627_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_5628_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_5629_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_5630_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_5631_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
     => ( ( distinct_real @ Xs2 )
       => ( ( linord4252657396651189596t_real @ ( set_real2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5632_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( ( distinct_o @ Xs2 )
       => ( ( linord3142498349692569832_set_o @ ( set_o2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5633_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( ( distinct_rat @ Xs2 )
       => ( ( linord1979837681955606664et_rat @ ( set_rat2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5634_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( ( distinct_num @ Xs2 )
       => ( ( linord8395671565052656842et_num @ ( set_num2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5635_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( ( distinct_int @ Xs2 )
       => ( ( linord2612477271533052124et_int @ ( set_int2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5636_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( ( distinct_nat @ Xs2 )
       => ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs2 ) )
          = Xs2 ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_5637_sgn__zero,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_zero
thf(fact_5638_sgn__zero,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_zero
thf(fact_5639_lemma__interval,axiom,
    ! [A: real,X4: real,B: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D2 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_5640_sin__bound__lemma,axiom,
    ! [X4: real,Y: real,U: real,V: real] :
      ( ( X4 = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_5641_lemma__interval__lt,axiom,
    ! [A: real,X4: real,B: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y5 ) ) @ D2 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_5642_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X4: real,Y: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ X4 @ Y ) )
      = ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( sgn_sgn_real @ Y ) ) ) ).

% Real_Vector_Spaces.sgn_mult
thf(fact_5643_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X4: complex,Y: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ X4 @ Y ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ X4 ) @ ( sgn_sgn_complex @ Y ) ) ) ).

% Real_Vector_Spaces.sgn_mult
thf(fact_5644_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( linord2324613341767563021nteger @ ( insert_Code_integer @ X4 @ A3 ) )
        = ( linord7786967269120229815nteger
          @ ^ [X: code_integer] : X
          @ X4
          @ ( linord2324613341767563021nteger @ ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5645_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( linord4252657396651189596t_real @ ( insert_real @ X4 @ A3 ) )
        = ( linord1674302359176591317l_real
          @ ^ [X: real] : X
          @ X4
          @ ( linord4252657396651189596t_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5646_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( linord3142498349692569832_set_o @ ( insert_o @ X4 @ A3 ) )
        = ( linord5141348845282165115ey_o_o
          @ ^ [X: $o] : X
          @ X4
          @ ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5647_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( linord2612477271533052124et_int @ ( insert_int @ X4 @ A3 ) )
        = ( linord734827384618529109nt_int
          @ ^ [X: int] : X
          @ X4
          @ ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5648_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A3 ) )
        = ( linord8961336180081300637at_nat
          @ ^ [X: nat] : X
          @ X4
          @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5649_sgn__zero__iff,axiom,
    ! [X4: complex] :
      ( ( ( sgn_sgn_complex @ X4 )
        = zero_zero_complex )
      = ( X4 = zero_zero_complex ) ) ).

% sgn_zero_iff
thf(fact_5650_sgn__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( sgn_sgn_real @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% sgn_zero_iff
thf(fact_5651_sgn__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_5652_zero__le__sgn__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_sgn_iff
thf(fact_5653_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X4 @ A3 )
       => ( ( linord3142498349692569832_set_o @ ( insert_o @ X4 @ A3 ) )
          = ( linord5141348845282165115ey_o_o
            @ ^ [X: $o] : X
            @ X4
            @ ( linord3142498349692569832_set_o @ A3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5654_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ~ ( member_real @ X4 @ A3 )
       => ( ( linord4252657396651189596t_real @ ( insert_real @ X4 @ A3 ) )
          = ( linord1674302359176591317l_real
            @ ^ [X: real] : X
            @ X4
            @ ( linord4252657396651189596t_real @ A3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5655_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ~ ( member_int @ X4 @ A3 )
       => ( ( linord2612477271533052124et_int @ ( insert_int @ X4 @ A3 ) )
          = ( linord734827384618529109nt_int
            @ ^ [X: int] : X
            @ X4
            @ ( linord2612477271533052124et_int @ A3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5656_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ~ ( member_Code_integer @ X4 @ A3 )
       => ( ( linord2324613341767563021nteger @ ( insert_Code_integer @ X4 @ A3 ) )
          = ( linord7786967269120229815nteger
            @ ^ [X: code_integer] : X
            @ X4
            @ ( linord2324613341767563021nteger @ A3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5657_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ X4 @ A3 )
       => ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A3 ) )
          = ( linord8961336180081300637at_nat
            @ ^ [X: nat] : X
            @ X4
            @ ( linord2614967742042102400et_nat @ A3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5658_insort__key_Osimps_I2_J,axiom,
    ! [F: nat > rat,X4: nat,Y: nat,Ys: list_nat] :
      ( ( ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8326206119994804901at_rat @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ X4 @ ( cons_nat @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8326206119994804901at_rat @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ Y @ ( linord8326206119994804901at_rat @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5659_insort__key_Osimps_I2_J,axiom,
    ! [F: int > rat,X4: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord102187795041083649nt_rat @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X4 @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord102187795041083649nt_rat @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord102187795041083649nt_rat @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5660_insort__key_Osimps_I2_J,axiom,
    ! [F: nat > num,X4: nat,Y: nat,Ys: list_nat] :
      ( ( ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord5518667966237079271at_num @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ X4 @ ( cons_nat @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord5518667966237079271at_num @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ Y @ ( linord5518667966237079271at_num @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5661_insort__key_Osimps_I2_J,axiom,
    ! [F: int > num,X4: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord6518021678138133827nt_num @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X4 @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord6518021678138133827nt_num @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord6518021678138133827nt_num @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5662_insort__key_Osimps_I2_J,axiom,
    ! [F: nat > nat,X4: nat,Y: nat,Ys: list_nat] :
      ( ( ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8961336180081300637at_nat @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ X4 @ ( cons_nat @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8961336180081300637at_nat @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ Y @ ( linord8961336180081300637at_nat @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5663_insort__key_Osimps_I2_J,axiom,
    ! [F: int > nat,X4: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord737317855127579385nt_nat @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X4 @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord737317855127579385nt_nat @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord737317855127579385nt_nat @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5664_insort__key_Osimps_I2_J,axiom,
    ! [F: nat > int,X4: nat,Y: nat,Ys: list_nat] :
      ( ( ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8958845709572250361at_int @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ X4 @ ( cons_nat @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord8958845709572250361at_int @ F @ X4 @ ( cons_nat @ Y @ Ys ) )
          = ( cons_nat @ Y @ ( linord8958845709572250361at_int @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5665_insort__key_Osimps_I2_J,axiom,
    ! [F: int > int,X4: int,Y: int,Ys: list_int] :
      ( ( ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord734827384618529109nt_int @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ X4 @ ( cons_int @ Y @ Ys ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y ) )
       => ( ( linord734827384618529109nt_int @ F @ X4 @ ( cons_int @ Y @ Ys ) )
          = ( cons_int @ Y @ ( linord734827384618529109nt_int @ F @ X4 @ Ys ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_5666_sorted__insort,axiom,
    ! [X4: rat,Xs2: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat
        @ ( linord8009651476059938989at_rat
          @ ^ [X: rat] : X
          @ X4
          @ Xs2 ) )
      = ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 ) ) ).

% sorted_insort
thf(fact_5667_sorted__insort,axiom,
    ! [X4: num,Xs2: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num
        @ ( linord3648998369544548657um_num
          @ ^ [X: num] : X
          @ X4
          @ Xs2 ) )
      = ( sorted_wrt_num @ ord_less_eq_num @ Xs2 ) ) ).

% sorted_insort
thf(fact_5668_sorted__insort,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat
        @ ( linord8961336180081300637at_nat
          @ ^ [X: nat] : X
          @ X4
          @ Xs2 ) )
      = ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).

% sorted_insort
thf(fact_5669_sorted__insort,axiom,
    ! [X4: int,Xs2: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int
        @ ( linord734827384618529109nt_int
          @ ^ [X: int] : X
          @ X4
          @ Xs2 ) )
      = ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).

% sorted_insort
thf(fact_5670_insort__is__Cons,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > rat,A: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord2597477761626359633BT_rat @ F @ A @ Xs2 )
        = ( cons_VEBT_VEBT @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5671_insort__is__Cons,axiom,
    ! [Xs2: list_real,F: real > rat,A: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord5462632957955297153al_rat @ F @ A @ Xs2 )
        = ( cons_real @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5672_insort__is__Cons,axiom,
    ! [Xs2: list_o,F: $o > rat,A: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord2703314976885039797_o_rat @ F @ A @ Xs2 )
        = ( cons_o @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5673_insort__is__Cons,axiom,
    ! [Xs2: list_nat,F: nat > rat,A: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord8326206119994804901at_rat @ F @ A @ Xs2 )
        = ( cons_nat @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5674_insort__is__Cons,axiom,
    ! [Xs2: list_int,F: int > rat,A: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord102187795041083649nt_rat @ F @ A @ Xs2 )
        = ( cons_int @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5675_insort__is__Cons,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > num,A: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_num @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord9013311644723409811BT_num @ F @ A @ Xs2 )
        = ( cons_VEBT_VEBT @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5676_insort__is__Cons,axiom,
    ! [Xs2: list_real,F: real > num,A: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_num @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord2655094804197571523al_num @ F @ A @ Xs2 )
        = ( cons_real @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5677_insort__is__Cons,axiom,
    ! [Xs2: list_o,F: $o > num,A: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_num @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord9119148859982089975_o_num @ F @ A @ Xs2 )
        = ( cons_o @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5678_insort__is__Cons,axiom,
    ! [Xs2: list_nat,F: nat > num,A: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_num @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord5518667966237079271at_num @ F @ A @ Xs2 )
        = ( cons_nat @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5679_insort__is__Cons,axiom,
    ! [Xs2: list_int,F: int > num,A: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_num @ ( F @ A ) @ ( F @ X3 ) ) )
     => ( ( linord6518021678138133827nt_num @ F @ A @ Xs2 )
        = ( cons_int @ A @ Xs2 ) ) ) ).

% insort_is_Cons
thf(fact_5680_sorted__insort__key,axiom,
    ! [F: vEBT_VEBT > real,X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_VEBT_VEBT_real @ F @ ( linord210457350419892133T_real @ F @ X4 @ Xs2 ) ) )
      = ( sorted_wrt_real @ ord_less_eq_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) ) ) ).

% sorted_insort_key
thf(fact_5681_sorted__insort__key,axiom,
    ! [F: product_prod_o_o > $o,X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( map_Pr7541730621154948341_o_o_o @ F @ ( linord6845329431107887148_o_o_o @ F @ X4 @ Xs2 ) ) )
      = ( sorted_wrt_o @ ord_less_eq_o @ ( map_Pr7541730621154948341_o_o_o @ F @ Xs2 ) ) ) ).

% sorted_insort_key
thf(fact_5682_sorted__insort__key,axiom,
    ! [F: nat > $o,X4: nat,Xs2: list_nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ ( map_nat_o @ F @ ( linord6441116398079720843_nat_o @ F @ X4 @ Xs2 ) ) )
      = ( sorted_wrt_o @ ord_less_eq_o @ ( map_nat_o @ F @ Xs2 ) ) ) ).

% sorted_insort_key
thf(fact_5683_sorted__insort__key,axiom,
    ! [F: vEBT_VEBT > nat,X4: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_VEBT_VEBT_nat @ F @ ( linord3232607821712855369BT_nat @ F @ X4 @ Xs2 ) ) )
      = ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) ) ) ).

% sorted_insort_key
thf(fact_5684_sorted__insort__key,axiom,
    ! [F: nat > nat,X4: nat,Xs2: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X4 @ Xs2 ) ) )
      = ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).

% sorted_insort_key
thf(fact_5685_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_5686_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_5687_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_5688_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_5689_mult__commute__abs,axiom,
    ! [C: assn] :
      ( ( ^ [X: assn] : ( times_times_assn @ X @ C ) )
      = ( times_times_assn @ C ) ) ).

% mult_commute_abs
thf(fact_5690_mult__commute__abs,axiom,
    ! [C: complex] :
      ( ( ^ [X: complex] : ( times_times_complex @ X @ C ) )
      = ( times_times_complex @ C ) ) ).

% mult_commute_abs
thf(fact_5691_insort__remove1,axiom,
    ! [A: real,Xs2: list_real] :
      ( ( member_real @ A @ ( set_real2 @ Xs2 ) )
     => ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
       => ( ( linord1674302359176591317l_real
            @ ^ [X: real] : X
            @ A
            @ ( remove1_real @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5692_insort__remove1,axiom,
    ! [A: $o,Xs2: list_o] :
      ( ( member_o @ A @ ( set_o2 @ Xs2 ) )
     => ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
       => ( ( linord5141348845282165115ey_o_o
            @ ^ [X: $o] : X
            @ A
            @ ( remove1_o @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5693_insort__remove1,axiom,
    ! [A: rat,Xs2: list_rat] :
      ( ( member_rat @ A @ ( set_rat2 @ Xs2 ) )
     => ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
       => ( ( linord8009651476059938989at_rat
            @ ^ [X: rat] : X
            @ A
            @ ( remove1_rat @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5694_insort__remove1,axiom,
    ! [A: num,Xs2: list_num] :
      ( ( member_num @ A @ ( set_num2 @ Xs2 ) )
     => ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
       => ( ( linord3648998369544548657um_num
            @ ^ [X: num] : X
            @ A
            @ ( remove1_num @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5695_insort__remove1,axiom,
    ! [A: nat,Xs2: list_nat] :
      ( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
       => ( ( linord8961336180081300637at_nat
            @ ^ [X: nat] : X
            @ A
            @ ( remove1_nat @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5696_insort__remove1,axiom,
    ! [A: int,Xs2: list_int] :
      ( ( member_int @ A @ ( set_int2 @ Xs2 ) )
     => ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
       => ( ( linord734827384618529109nt_int
            @ ^ [X: int] : X
            @ A
            @ ( remove1_int @ A @ Xs2 ) )
          = Xs2 ) ) ) ).

% insort_remove1
thf(fact_5697_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ X4 @ A3 )
       => ( ( linord2324613341767563021nteger @ A3 )
          = ( linord7786967269120229815nteger
            @ ^ [X: code_integer] : X
            @ X4
            @ ( linord2324613341767563021nteger @ ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_5698_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X4 @ A3 )
       => ( ( linord4252657396651189596t_real @ A3 )
          = ( linord1674302359176591317l_real
            @ ^ [X: real] : X
            @ X4
            @ ( linord4252657396651189596t_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_5699_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X4 @ A3 )
       => ( ( linord3142498349692569832_set_o @ A3 )
          = ( linord5141348845282165115ey_o_o
            @ ^ [X: $o] : X
            @ X4
            @ ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_5700_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X4 @ A3 )
       => ( ( linord2612477271533052124et_int @ A3 )
          = ( linord734827384618529109nt_int
            @ ^ [X: int] : X
            @ X4
            @ ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_5701_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X4 @ A3 )
       => ( ( linord2614967742042102400et_nat @ A3 )
          = ( linord8961336180081300637at_nat
            @ ^ [X: nat] : X
            @ X4
            @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_5702_sorted__list__of__set__def,axiom,
    ( linord2614967742042102400et_nat
    = ( linord1089935798310486446at_nat
      @ ^ [X: nat] : X ) ) ).

% sorted_list_of_set_def
thf(fact_5703_fun__upd__None__restrict,axiom,
    ! [X4: vEBT_VEBT,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_nat] :
      ( ( ( member_VEBT_VEBT @ X4 @ D4 )
       => ( ( fun_up5885881570350532375on_nat @ ( restri774867724463461460BT_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restri774867724463461460BT_nat @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ D4 )
       => ( ( fun_up5885881570350532375on_nat @ ( restri774867724463461460BT_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restri774867724463461460BT_nat @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5704_fun__upd__None__restrict,axiom,
    ! [X4: vEBT_VEBT,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_num] :
      ( ( ( member_VEBT_VEBT @ X4 @ D4 )
       => ( ( fun_up6594125507299370081on_num @ ( restri6555571547474015902BT_num @ M @ D4 ) @ X4 @ none_num )
          = ( restri6555571547474015902BT_num @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ D4 )
       => ( ( fun_up6594125507299370081on_num @ ( restri6555571547474015902BT_num @ M @ D4 ) @ X4 @ none_num )
          = ( restri6555571547474015902BT_num @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5705_fun__upd__None__restrict,axiom,
    ! [X4: real,D4: set_real,M: real > option_nat] :
      ( ( ( member_real @ X4 @ D4 )
       => ( ( fun_up6677080212936659659on_nat @ ( restri6827137924477938990al_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restri6827137924477938990al_nat @ M @ ( minus_minus_set_real @ D4 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) )
      & ( ~ ( member_real @ X4 @ D4 )
       => ( ( fun_up6677080212936659659on_nat @ ( restri6827137924477938990al_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restri6827137924477938990al_nat @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5706_fun__upd__None__restrict,axiom,
    ! [X4: real,D4: set_real,M: real > option_num] :
      ( ( ( member_real @ X4 @ D4 )
       => ( ( fun_up7385324149885497365on_num @ ( restri3384469710633717624al_num @ M @ D4 ) @ X4 @ none_num )
          = ( restri3384469710633717624al_num @ M @ ( minus_minus_set_real @ D4 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) )
      & ( ~ ( member_real @ X4 @ D4 )
       => ( ( fun_up7385324149885497365on_num @ ( restri3384469710633717624al_num @ M @ D4 ) @ X4 @ none_num )
          = ( restri3384469710633717624al_num @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5707_fun__upd__None__restrict,axiom,
    ! [X4: $o,D4: set_o,M: $o > option_nat] :
      ( ( ( member_o @ X4 @ D4 )
       => ( ( fun_upd_o_option_nat @ ( restrict_map_o_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_o_nat @ M @ ( minus_minus_set_o @ D4 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) )
      & ( ~ ( member_o @ X4 @ D4 )
       => ( ( fun_upd_o_option_nat @ ( restrict_map_o_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_o_nat @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5708_fun__upd__None__restrict,axiom,
    ! [X4: $o,D4: set_o,M: $o > option_num] :
      ( ( ( member_o @ X4 @ D4 )
       => ( ( fun_upd_o_option_num @ ( restrict_map_o_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_o_num @ M @ ( minus_minus_set_o @ D4 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) )
      & ( ~ ( member_o @ X4 @ D4 )
       => ( ( fun_upd_o_option_num @ ( restrict_map_o_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_o_num @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5709_fun__upd__None__restrict,axiom,
    ! [X4: int,D4: set_int,M: int > option_nat] :
      ( ( ( member_int @ X4 @ D4 )
       => ( ( fun_up3620524117960394059on_nat @ ( restrict_map_int_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_int_nat @ M @ ( minus_minus_set_int @ D4 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) )
      & ( ~ ( member_int @ X4 @ D4 )
       => ( ( fun_up3620524117960394059on_nat @ ( restrict_map_int_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_int_nat @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5710_fun__upd__None__restrict,axiom,
    ! [X4: int,D4: set_int,M: int > option_num] :
      ( ( ( member_int @ X4 @ D4 )
       => ( ( fun_up4328768054909231765on_num @ ( restrict_map_int_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_int_num @ M @ ( minus_minus_set_int @ D4 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) )
      & ( ~ ( member_int @ X4 @ D4 )
       => ( ( fun_up4328768054909231765on_num @ ( restrict_map_int_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_int_num @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5711_fun__upd__None__restrict,axiom,
    ! [X4: nat,D4: set_nat,M: nat > option_nat] :
      ( ( ( member_nat @ X4 @ D4 )
       => ( ( fun_up1493157387958331631on_nat @ ( restrict_map_nat_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_nat_nat @ M @ ( minus_minus_set_nat @ D4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X4 @ D4 )
       => ( ( fun_up1493157387958331631on_nat @ ( restrict_map_nat_nat @ M @ D4 ) @ X4 @ none_nat )
          = ( restrict_map_nat_nat @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5712_fun__upd__None__restrict,axiom,
    ! [X4: nat,D4: set_nat,M: nat > option_num] :
      ( ( ( member_nat @ X4 @ D4 )
       => ( ( fun_up2201401324907169337on_num @ ( restrict_map_nat_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_nat_num @ M @ ( minus_minus_set_nat @ D4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X4 @ D4 )
       => ( ( fun_up2201401324907169337on_num @ ( restrict_map_nat_num @ M @ D4 ) @ X4 @ none_num )
          = ( restrict_map_nat_num @ M @ D4 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_5713_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X4: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X4 @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X4 @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X4 @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_5714_total__on__singleton,axiom,
    ! [X4: vEBT_VEBT] : ( total_on_VEBT_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) @ ( insert494605675473494903T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ X4 ) @ bot_bo5088076668136028147T_VEBT ) ) ).

% total_on_singleton
thf(fact_5715_total__on__singleton,axiom,
    ! [X4: code_integer] : ( total_8516208958838685657nteger @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) @ ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ X4 @ X4 ) @ bot_bo4276436098303576167nteger ) ) ).

% total_on_singleton
thf(fact_5716_total__on__singleton,axiom,
    ! [X4: real] : ( total_on_real @ ( insert_real @ X4 @ bot_bot_set_real ) @ ( insert7746734233410687241l_real @ ( produc4511245868158468465l_real @ X4 @ X4 ) @ bot_bo3948376660626123781l_real ) ) ).

% total_on_singleton
thf(fact_5717_total__on__singleton,axiom,
    ! [X4: $o] : ( total_on_o @ ( insert_o @ X4 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ X4 ) @ bot_bo7073875226086086771od_o_o ) ) ).

% total_on_singleton
thf(fact_5718_total__on__singleton,axiom,
    ! [X4: nat] : ( total_on_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ bot_bo2099793752762293965at_nat ) ) ).

% total_on_singleton
thf(fact_5719_total__on__singleton,axiom,
    ! [X4: int] : ( total_on_int @ ( insert_int @ X4 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ bot_bo1796632182523588997nt_int ) ) ).

% total_on_singleton
thf(fact_5720_nth__step__trancl,axiom,
    ! [Xs2: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ one_one_nat ) )
         => ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ ( suc @ N2 ) ) @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member660371905731732212_VEBTi @ ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N ) @ ( nth_VEBT_VEBTi @ Xs2 @ M ) ) @ ( transi2803566869205510612_VEBTi @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5721_nth__step__trancl,axiom,
    ! [Xs2: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ one_one_nat ) )
         => ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( suc @ N2 ) ) @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( nth_VEBT_VEBT @ Xs2 @ M ) ) @ ( transi8906537157094044885T_VEBT @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5722_nth__step__trancl,axiom,
    ! [Xs2: list_Code_integer,R: set_Pr4811707699266497531nteger,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ one_one_nat ) )
         => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ ( suc @ N2 ) ) @ ( nth_Code_integer @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_s3445333598471063425nteger @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ N ) @ ( nth_Code_integer @ Xs2 @ M ) ) @ ( transi6870300401645067644nteger @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5723_nth__step__trancl,axiom,
    ! [Xs2: list_real,R: set_Pr6218003697084177305l_real,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_real @ Xs2 ) @ one_one_nat ) )
         => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs2 @ ( suc @ N2 ) ) @ ( nth_real @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ ( nth_real @ Xs2 @ N ) @ ( nth_real @ Xs2 @ M ) ) @ ( transi1789104906590519371l_real @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5724_nth__step__trancl,axiom,
    ! [Xs2: list_o,R: set_Product_prod_o_o,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat ) )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( suc @ N2 ) ) @ ( nth_o @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs2 @ N ) @ ( nth_o @ Xs2 @ M ) ) @ ( transitive_trancl_o @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5725_nth__step__trancl,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ ( suc @ N2 ) ) @ ( nth_nat @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ N ) @ ( nth_nat @ Xs2 @ M ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5726_nth__step__trancl,axiom,
    ! [Xs2: list_int,R: set_Pr958786334691620121nt_int,N: nat,M: nat] :
      ( ! [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat ) )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( suc @ N2 ) ) @ ( nth_int @ Xs2 @ N2 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( ord_less_nat @ M @ N )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs2 @ N ) @ ( nth_int @ Xs2 @ M ) ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5727_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ( nth_Pr6329974346453275474_VEBTi @ ( zip_VE793581609497812771_VEBTi @ Xs2 @ Ys ) @ I )
          = ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5728_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr8725177398587324397T_VEBT @ ( zip_VE7413257051550508102T_VEBT @ Xs2 @ Ys ) @ I )
          = ( produc7053807326796202854T_VEBT @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5729_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ( nth_Pr316670251186196177_VEBTi @ ( zip_VE6444338338598820466_VEBTi @ Xs2 @ Ys ) @ I )
          = ( produc6084888613844515218_VEBTi @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5730_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Ys ) @ I )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5731_nth__zip,axiom,
    ! [I: nat,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ Ys ) )
       => ( ( nth_Pr2304437835452373666nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) @ I )
          = ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ I ) @ ( nth_Code_integer @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5732_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,Ys: list_real] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Ys ) )
       => ( ( nth_Pr3433448822664029129i_real @ ( zip_VEBT_VEBTi_real @ Xs2 @ Ys ) @ I )
          = ( produc8457151488442208762i_real @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_real @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5733_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Ys ) )
       => ( ( nth_Pr6842391030413306568T_real @ ( zip_VEBT_VEBT_real @ Xs2 @ Ys ) @ I )
          = ( produc8117437818029410057T_real @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_real @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5734_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr3306050735993963089EBTi_o @ ( zip_VEBT_VEBTi_o @ Xs2 @ Ys ) @ I )
          = ( produc8194178580519725514EBTi_o @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5735_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs2 @ Ys ) @ I )
          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5736_nth__zip,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr6911489093701683181Ti_nat @ ( zip_VEBT_VEBTi_nat @ Xs2 @ Ys ) @ I )
          = ( produc7192665754729510430Ti_nat @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_5737_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_5738_prod__encode__eq,axiom,
    ! [X4: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_encode @ X4 )
        = ( nat_prod_encode @ Y ) )
      = ( X4 = Y ) ) ).

% prod_encode_eq
thf(fact_5739_image__ident,axiom,
    ! [Y7: set_nat] :
      ( ( image_nat_nat
        @ ^ [X: nat] : X
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_5740_image__ident,axiom,
    ! [Y7: set_int] :
      ( ( image_int_int
        @ ^ [X: int] : X
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_5741_image__is__empty,axiom,
    ! [F: real > real,A3: set_real] :
      ( ( ( image_real_real @ F @ A3 )
        = bot_bot_set_real )
      = ( A3 = bot_bot_set_real ) ) ).

% image_is_empty
thf(fact_5742_image__is__empty,axiom,
    ! [F: $o > real,A3: set_o] :
      ( ( ( image_o_real @ F @ A3 )
        = bot_bot_set_real )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5743_image__is__empty,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ( ( image_nat_real @ F @ A3 )
        = bot_bot_set_real )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5744_image__is__empty,axiom,
    ! [F: int > real,A3: set_int] :
      ( ( ( image_int_real @ F @ A3 )
        = bot_bot_set_real )
      = ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_5745_image__is__empty,axiom,
    ! [F: real > $o,A3: set_real] :
      ( ( ( image_real_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_real ) ) ).

% image_is_empty
thf(fact_5746_image__is__empty,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( ( image_o_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5747_image__is__empty,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( ( image_nat_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5748_image__is__empty,axiom,
    ! [F: int > $o,A3: set_int] :
      ( ( ( image_int_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_5749_image__is__empty,axiom,
    ! [F: real > nat,A3: set_real] :
      ( ( ( image_real_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_real ) ) ).

% image_is_empty
thf(fact_5750_image__is__empty,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( ( image_o_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5751_empty__is__image,axiom,
    ! [F: real > real,A3: set_real] :
      ( ( bot_bot_set_real
        = ( image_real_real @ F @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% empty_is_image
thf(fact_5752_empty__is__image,axiom,
    ! [F: $o > real,A3: set_o] :
      ( ( bot_bot_set_real
        = ( image_o_real @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5753_empty__is__image,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ( bot_bot_set_real
        = ( image_nat_real @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5754_empty__is__image,axiom,
    ! [F: int > real,A3: set_int] :
      ( ( bot_bot_set_real
        = ( image_int_real @ F @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_5755_empty__is__image,axiom,
    ! [F: real > $o,A3: set_real] :
      ( ( bot_bot_set_o
        = ( image_real_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% empty_is_image
thf(fact_5756_empty__is__image,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5757_empty__is__image,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5758_empty__is__image,axiom,
    ! [F: int > $o,A3: set_int] :
      ( ( bot_bot_set_o
        = ( image_int_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_5759_empty__is__image,axiom,
    ! [F: real > nat,A3: set_real] :
      ( ( bot_bot_set_nat
        = ( image_real_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% empty_is_image
thf(fact_5760_empty__is__image,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5761_image__empty,axiom,
    ! [F: real > real] :
      ( ( image_real_real @ F @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% image_empty
thf(fact_5762_image__empty,axiom,
    ! [F: real > $o] :
      ( ( image_real_o @ F @ bot_bot_set_real )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5763_image__empty,axiom,
    ! [F: real > nat] :
      ( ( image_real_nat @ F @ bot_bot_set_real )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_5764_image__empty,axiom,
    ! [F: real > int] :
      ( ( image_real_int @ F @ bot_bot_set_real )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_5765_image__empty,axiom,
    ! [F: $o > real] :
      ( ( image_o_real @ F @ bot_bot_set_o )
      = bot_bot_set_real ) ).

% image_empty
thf(fact_5766_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5767_image__empty,axiom,
    ! [F: $o > nat] :
      ( ( image_o_nat @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_5768_image__empty,axiom,
    ! [F: $o > int] :
      ( ( image_o_int @ F @ bot_bot_set_o )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_5769_image__empty,axiom,
    ! [F: nat > real] :
      ( ( image_nat_real @ F @ bot_bot_set_nat )
      = bot_bot_set_real ) ).

% image_empty
thf(fact_5770_image__empty,axiom,
    ! [F: nat > $o] :
      ( ( image_nat_o @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5771_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_real,A6: list_real,B6: list_real] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_real_real @ A @ B )
            = ( zip_real_real @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5772_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_o,A6: list_real,B6: list_o] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_real_o @ A @ B )
            = ( zip_real_o @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5773_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_nat,A6: list_real,B6: list_nat] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_real_nat @ A @ B )
            = ( zip_real_nat @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5774_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_int,A6: list_real,B6: list_int] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_real_int @ A @ B )
            = ( zip_real_int @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5775_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_real,A6: list_o,B6: list_real] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_o_real @ A @ B )
            = ( zip_o_real @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5776_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_o,A6: list_o,B6: list_o] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_o_o @ A @ B )
            = ( zip_o_o @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5777_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_nat,A6: list_o,B6: list_nat] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_o_nat @ A @ B )
            = ( zip_o_nat @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5778_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_int,A6: list_o,B6: list_int] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_o_int @ A @ B )
            = ( zip_o_int @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5779_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_real,A6: list_nat,B6: list_real] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_nat @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_nat_real @ A @ B )
            = ( zip_nat_real @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5780_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_o,A6: list_nat,B6: list_o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_nat @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_nat_o @ A @ B )
            = ( zip_nat_o @ A6 @ B6 ) )
          = ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5781_image__add__0,axiom,
    ! [S3: set_complex] :
      ( ( image_1468599708987790691omplex @ ( plus_plus_complex @ zero_zero_complex ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_5782_image__add__0,axiom,
    ! [S3: set_real] :
      ( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_5783_image__add__0,axiom,
    ! [S3: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_5784_image__add__0,axiom,
    ! [S3: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_5785_image__add__0,axiom,
    ! [S3: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_5786_restrict__map__to__empty,axiom,
    ! [M: real > option_nat] :
      ( ( restri6827137924477938990al_nat @ M @ bot_bot_set_real )
      = ( ^ [X: real] : none_nat ) ) ).

% restrict_map_to_empty
thf(fact_5787_restrict__map__to__empty,axiom,
    ! [M: real > option_num] :
      ( ( restri3384469710633717624al_num @ M @ bot_bot_set_real )
      = ( ^ [X: real] : none_num ) ) ).

% restrict_map_to_empty
thf(fact_5788_restrict__map__to__empty,axiom,
    ! [M: $o > option_nat] :
      ( ( restrict_map_o_nat @ M @ bot_bot_set_o )
      = ( ^ [X: $o] : none_nat ) ) ).

% restrict_map_to_empty
thf(fact_5789_restrict__map__to__empty,axiom,
    ! [M: $o > option_num] :
      ( ( restrict_map_o_num @ M @ bot_bot_set_o )
      = ( ^ [X: $o] : none_num ) ) ).

% restrict_map_to_empty
thf(fact_5790_restrict__map__to__empty,axiom,
    ! [M: nat > option_nat] :
      ( ( restrict_map_nat_nat @ M @ bot_bot_set_nat )
      = ( ^ [X: nat] : none_nat ) ) ).

% restrict_map_to_empty
thf(fact_5791_restrict__map__to__empty,axiom,
    ! [M: nat > option_num] :
      ( ( restrict_map_nat_num @ M @ bot_bot_set_nat )
      = ( ^ [X: nat] : none_num ) ) ).

% restrict_map_to_empty
thf(fact_5792_restrict__map__to__empty,axiom,
    ! [M: int > option_nat] :
      ( ( restrict_map_int_nat @ M @ bot_bot_set_int )
      = ( ^ [X: int] : none_nat ) ) ).

% restrict_map_to_empty
thf(fact_5793_restrict__map__to__empty,axiom,
    ! [M: int > option_num] :
      ( ( restrict_map_int_num @ M @ bot_bot_set_int )
      = ( ^ [X: int] : none_num ) ) ).

% restrict_map_to_empty
thf(fact_5794_restrict__map__to__empty,axiom,
    ! [M: real > option4927543243414619207at_nat] :
      ( ( restri4688977514882730531at_nat @ M @ bot_bot_set_real )
      = ( ^ [X: real] : none_P5556105721700978146at_nat ) ) ).

% restrict_map_to_empty
thf(fact_5795_restrict__map__to__empty,axiom,
    ! [M: $o > option4927543243414619207at_nat] :
      ( ( restri2754309951621987225at_nat @ M @ bot_bot_set_o )
      = ( ^ [X: $o] : none_P5556105721700978146at_nat ) ) ).

% restrict_map_to_empty
thf(fact_5796_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_5797_image__add__atLeastLessThan_H,axiom,
    ! [K: real,I: real,J: real] :
      ( ( image_real_real
        @ ^ [N4: real] : ( plus_plus_real @ N4 @ K )
        @ ( set_or66887138388493659n_real @ I @ J ) )
      = ( set_or66887138388493659n_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_5798_image__add__atLeastLessThan_H,axiom,
    ! [K: rat,I: rat,J: rat] :
      ( ( image_rat_rat
        @ ^ [N4: rat] : ( plus_plus_rat @ N4 @ K )
        @ ( set_or4029947393144176647an_rat @ I @ J ) )
      = ( set_or4029947393144176647an_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_5799_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat
        @ ^ [N4: nat] : ( plus_plus_nat @ N4 @ K )
        @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_5800_image__add__atLeastLessThan_H,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int
        @ ^ [N4: int] : ( plus_plus_int @ N4 @ K )
        @ ( set_or4662586982721622107an_int @ I @ J ) )
      = ( set_or4662586982721622107an_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_5801_image__add__atLeastLessThan_H,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [N4: code_integer] : ( plus_p5714425477246183910nteger @ N4 @ K )
        @ ( set_or8404916559141939852nteger @ I @ J ) )
      = ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_5802_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_5803_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_5804_trancl__single,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( transi6870300401645067644nteger @ ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ A @ B ) @ bot_bo4276436098303576167nteger ) )
      = ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ A @ B ) @ bot_bo4276436098303576167nteger ) ) ).

% trancl_single
thf(fact_5805_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_5806_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_5807_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_5808_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_5809_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_5810_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_5811_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_5812_imageE,axiom,
    ! [B: nat,F: nat > nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5813_imageE,axiom,
    ! [B: nat,F: vEBT_VEBT > nat,A3: set_VEBT_VEBT] :
      ( ( member_nat @ B @ ( image_VEBT_VEBT_nat @ F @ A3 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5814_imageE,axiom,
    ! [B: nat,F: real > nat,A3: set_real] :
      ( ( member_nat @ B @ ( image_real_nat @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5815_imageE,axiom,
    ! [B: nat,F: int > nat,A3: set_int] :
      ( ( member_nat @ B @ ( image_int_nat @ F @ A3 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5816_imageE,axiom,
    ! [B: vEBT_VEBT,F: nat > vEBT_VEBT,A3: set_nat] :
      ( ( member_VEBT_VEBT @ B @ ( image_nat_VEBT_VEBT @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5817_imageE,axiom,
    ! [B: vEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( image_3375948659692109573T_VEBT @ F @ A3 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5818_imageE,axiom,
    ! [B: vEBT_VEBT,F: real > vEBT_VEBT,A3: set_real] :
      ( ( member_VEBT_VEBT @ B @ ( image_real_VEBT_VEBT @ F @ A3 ) )
     => ~ ! [X3: real] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_real @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5819_imageE,axiom,
    ! [B: vEBT_VEBT,F: int > vEBT_VEBT,A3: set_int] :
      ( ( member_VEBT_VEBT @ B @ ( image_int_VEBT_VEBT @ F @ A3 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5820_imageE,axiom,
    ! [B: real,F: nat > real,A3: set_nat] :
      ( ( member_real @ B @ ( image_nat_real @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5821_imageE,axiom,
    ! [B: real,F: vEBT_VEBT > real,A3: set_VEBT_VEBT] :
      ( ( member_real @ B @ ( image_VEBT_VEBT_real @ F @ A3 ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_VEBT_VEBT @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5822_image__image,axiom,
    ! [F: real > real,G: nat > real,A3: set_nat] :
      ( ( image_real_real @ F @ ( image_nat_real @ G @ A3 ) )
      = ( image_nat_real
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5823_image__image,axiom,
    ! [F: real > nat,G: nat > real,A3: set_nat] :
      ( ( image_real_nat @ F @ ( image_nat_real @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5824_image__image,axiom,
    ! [F: real > int,G: nat > real,A3: set_nat] :
      ( ( image_real_int @ F @ ( image_nat_real @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5825_image__image,axiom,
    ! [F: int > real,G: nat > int,A3: set_nat] :
      ( ( image_int_real @ F @ ( image_nat_int @ G @ A3 ) )
      = ( image_nat_real
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5826_image__image,axiom,
    ! [F: int > nat,G: nat > int,A3: set_nat] :
      ( ( image_int_nat @ F @ ( image_nat_int @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5827_image__image,axiom,
    ! [F: nat > real,G: nat > nat,A3: set_nat] :
      ( ( image_nat_real @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_real
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5828_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5829_image__image,axiom,
    ! [F: nat > int,G: int > nat,A3: set_int] :
      ( ( image_nat_int @ F @ ( image_int_nat @ G @ A3 ) )
      = ( image_int_int
        @ ^ [X: int] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5830_image__image,axiom,
    ! [F: nat > int,G: nat > nat,A3: set_nat] :
      ( ( image_nat_int @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5831_image__image,axiom,
    ! [F: int > int,G: nat > int,A3: set_nat] :
      ( ( image_int_int @ F @ ( image_nat_int @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_5832_Compr__image__eq,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,A3: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( image_3375948659692109573T_VEBT @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_3375948659692109573T_VEBT @ F
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5833_Compr__image__eq,axiom,
    ! [F: real > vEBT_VEBT,A3: set_real,P: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( image_real_VEBT_VEBT @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_real_VEBT_VEBT @ F
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5834_Compr__image__eq,axiom,
    ! [F: vEBT_VEBT > real,A3: set_VEBT_VEBT,P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( image_VEBT_VEBT_real @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_VEBT_VEBT_real @ F
        @ ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5835_Compr__image__eq,axiom,
    ! [F: real > real,A3: set_real,P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( image_real_real @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_real_real @ F
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5836_Compr__image__eq,axiom,
    ! [F: complex > vEBT_VEBT,A3: set_complex,P: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( image_932796090930683071T_VEBT @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_932796090930683071T_VEBT @ F
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5837_Compr__image__eq,axiom,
    ! [F: complex > real,A3: set_complex,P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( image_complex_real @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_complex_real @ F
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5838_Compr__image__eq,axiom,
    ! [F: nat > vEBT_VEBT,A3: set_nat,P: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( image_nat_VEBT_VEBT @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_VEBT_VEBT @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5839_Compr__image__eq,axiom,
    ! [F: nat > real,A3: set_nat,P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( image_nat_real @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_real @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5840_Compr__image__eq,axiom,
    ! [F: int > vEBT_VEBT,A3: set_int,P: vEBT_VEBT > $o] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( image_int_VEBT_VEBT @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_int_VEBT_VEBT @ F
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5841_Compr__image__eq,axiom,
    ! [F: int > real,A3: set_int,P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( image_int_real @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image_int_real @ F
        @ ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5842_zip__inj,axiom,
    ! [A: list_real,B: list_real,A6: list_real,B6: list_real] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_real_real @ A @ B )
            = ( zip_real_real @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5843_zip__inj,axiom,
    ! [A: list_real,B: list_o,A6: list_real,B6: list_o] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_real_o @ A @ B )
            = ( zip_real_o @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5844_zip__inj,axiom,
    ! [A: list_real,B: list_nat,A6: list_real,B6: list_nat] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_real_nat @ A @ B )
            = ( zip_real_nat @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5845_zip__inj,axiom,
    ! [A: list_real,B: list_int,A6: list_real,B6: list_int] :
      ( ( ( size_size_list_real @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_real @ A6 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_real_int @ A @ B )
            = ( zip_real_int @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5846_zip__inj,axiom,
    ! [A: list_o,B: list_real,A6: list_o,B6: list_real] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_o_real @ A @ B )
            = ( zip_o_real @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5847_zip__inj,axiom,
    ! [A: list_o,B: list_o,A6: list_o,B6: list_o] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_o_o @ A @ B )
            = ( zip_o_o @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5848_zip__inj,axiom,
    ! [A: list_o,B: list_nat,A6: list_o,B6: list_nat] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_o_nat @ A @ B )
            = ( zip_o_nat @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5849_zip__inj,axiom,
    ! [A: list_o,B: list_int,A6: list_o,B6: list_int] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_o @ A6 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_o_int @ A @ B )
            = ( zip_o_int @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5850_zip__inj,axiom,
    ! [A: list_nat,B: list_real,A6: list_nat,B6: list_real] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_real @ B ) )
     => ( ( ( size_size_list_nat @ A6 )
          = ( size_size_list_real @ B6 ) )
       => ( ( ( zip_nat_real @ A @ B )
            = ( zip_nat_real @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5851_zip__inj,axiom,
    ! [A: list_nat,B: list_o,A6: list_nat,B6: list_o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_nat @ A6 )
          = ( size_size_list_o @ B6 ) )
       => ( ( ( zip_nat_o @ A @ B )
            = ( zip_nat_o @ A6 @ B6 ) )
         => ( ( A = A6 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5852_all__subset__image,axiom,
    ! [F: nat > real,A3: set_nat,P: set_real > $o] :
      ( ( ! [B5: set_real] :
            ( ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A3 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ord_less_eq_set_nat @ B5 @ A3 )
           => ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_5853_all__subset__image,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_set_nat > $o] :
      ( ( ! [B5: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F @ A3 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ord_less_eq_set_nat @ B5 @ A3 )
           => ( P @ ( image_nat_set_nat @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_5854_all__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ord_less_eq_set_nat @ B5 @ A3 )
           => ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_5855_all__subset__image,axiom,
    ! [F: nat > int,A3: set_nat,P: set_int > $o] :
      ( ( ! [B5: set_int] :
            ( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A3 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ord_less_eq_set_nat @ B5 @ A3 )
           => ( P @ ( image_nat_int @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_5856_all__subset__image,axiom,
    ! [F: int > int,A3: set_int,P: set_int > $o] :
      ( ( ! [B5: set_int] :
            ( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A3 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_int] :
            ( ( ord_less_eq_set_int @ B5 @ A3 )
           => ( P @ ( image_int_int @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_5857_subset__image__iff,axiom,
    ! [B4: set_real,F: nat > real,A3: set_nat] :
      ( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B4
              = ( image_nat_real @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5858_subset__image__iff,axiom,
    ! [B4: set_set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B4
              = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5859_subset__image__iff,axiom,
    ! [B4: set_nat,F: nat > nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B4
              = ( image_nat_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5860_subset__image__iff,axiom,
    ! [B4: set_int,F: nat > int,A3: set_nat] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B4
              = ( image_nat_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5861_subset__image__iff,axiom,
    ! [B4: set_int,F: int > int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A3 ) )
      = ( ? [AA: set_int] :
            ( ( ord_less_eq_set_int @ AA @ A3 )
            & ( B4
              = ( image_int_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5862_image__subset__iff,axiom,
    ! [F: nat > nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_nat @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_5863_image__subset__iff,axiom,
    ! [F: nat > real,A3: set_nat,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ A3 ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_real @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_5864_image__subset__iff,axiom,
    ! [F: nat > set_nat,A3: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_set_nat @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_5865_image__subset__iff,axiom,
    ! [F: nat > int,A3: set_nat,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_int @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_5866_image__subset__iff,axiom,
    ! [F: int > int,A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ B4 )
      = ( ! [X: int] :
            ( ( member_int @ X @ A3 )
           => ( member_int @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_5867_subset__imageE,axiom,
    ! [B4: set_real,F: nat > real,A3: set_nat] :
      ( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A3 ) )
     => ~ ! [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
           => ( B4
             != ( image_nat_real @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_5868_subset__imageE,axiom,
    ! [B4: set_set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A3 ) )
     => ~ ! [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
           => ( B4
             != ( image_nat_set_nat @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_5869_subset__imageE,axiom,
    ! [B4: set_nat,F: nat > nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
     => ~ ! [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
           => ( B4
             != ( image_nat_nat @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_5870_subset__imageE,axiom,
    ! [B4: set_int,F: nat > int,A3: set_nat] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_nat_int @ F @ A3 ) )
     => ~ ! [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
           => ( B4
             != ( image_nat_int @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_5871_subset__imageE,axiom,
    ! [B4: set_int,F: int > int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B4 @ ( image_int_int @ F @ A3 ) )
     => ~ ! [C6: set_int] :
            ( ( ord_less_eq_set_int @ C6 @ A3 )
           => ( B4
             != ( image_int_int @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_5872_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5873_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_nat_VEBT_VEBT @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5874_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > real,B4: set_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_nat_real @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5875_image__subsetI,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > nat,B4: set_nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_VEBT_VEBT_nat @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5876_image__subsetI,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_3375948659692109573T_VEBT @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5877_image__subsetI,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > real,B4: set_real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_VEBT_VEBT_real @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5878_image__subsetI,axiom,
    ! [A3: set_real,F: real > nat,B4: set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5879_image__subsetI,axiom,
    ! [A3: set_real,F: real > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_real_VEBT_VEBT @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5880_image__subsetI,axiom,
    ! [A3: set_real,F: real > real,B4: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_real_real @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5881_image__subsetI,axiom,
    ! [A3: set_int,F: int > nat,B4: set_nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A3 ) @ B4 ) ) ).

% image_subsetI
thf(fact_5882_image__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > real] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ord_less_eq_set_real @ ( image_nat_real @ F @ A3 ) @ ( image_nat_real @ F @ B4 ) ) ) ).

% image_mono
thf(fact_5883_image__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_5884_image__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_5885_image__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > int] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B4 ) ) ) ).

% image_mono
thf(fact_5886_image__mono,axiom,
    ! [A3: set_int,B4: set_int,F: int > int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B4 ) ) ) ).

% image_mono
thf(fact_5887_restrict__map__eq_I2_J,axiom,
    ! [M: nat > option_nat,A3: set_nat,K: nat,V: nat] :
      ( ( ( restrict_map_nat_nat @ M @ A3 @ K )
        = ( some_nat @ V ) )
      = ( ( ( M @ K )
          = ( some_nat @ V ) )
        & ( member_nat @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5888_restrict__map__eq_I2_J,axiom,
    ! [M: vEBT_VEBT > option_nat,A3: set_VEBT_VEBT,K: vEBT_VEBT,V: nat] :
      ( ( ( restri774867724463461460BT_nat @ M @ A3 @ K )
        = ( some_nat @ V ) )
      = ( ( ( M @ K )
          = ( some_nat @ V ) )
        & ( member_VEBT_VEBT @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5889_restrict__map__eq_I2_J,axiom,
    ! [M: real > option_nat,A3: set_real,K: real,V: nat] :
      ( ( ( restri6827137924477938990al_nat @ M @ A3 @ K )
        = ( some_nat @ V ) )
      = ( ( ( M @ K )
          = ( some_nat @ V ) )
        & ( member_real @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5890_restrict__map__eq_I2_J,axiom,
    ! [M: int > option_nat,A3: set_int,K: int,V: nat] :
      ( ( ( restrict_map_int_nat @ M @ A3 @ K )
        = ( some_nat @ V ) )
      = ( ( ( M @ K )
          = ( some_nat @ V ) )
        & ( member_int @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5891_restrict__map__eq_I2_J,axiom,
    ! [M: nat > option_num,A3: set_nat,K: nat,V: num] :
      ( ( ( restrict_map_nat_num @ M @ A3 @ K )
        = ( some_num @ V ) )
      = ( ( ( M @ K )
          = ( some_num @ V ) )
        & ( member_nat @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5892_restrict__map__eq_I2_J,axiom,
    ! [M: vEBT_VEBT > option_num,A3: set_VEBT_VEBT,K: vEBT_VEBT,V: num] :
      ( ( ( restri6555571547474015902BT_num @ M @ A3 @ K )
        = ( some_num @ V ) )
      = ( ( ( M @ K )
          = ( some_num @ V ) )
        & ( member_VEBT_VEBT @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5893_restrict__map__eq_I2_J,axiom,
    ! [M: real > option_num,A3: set_real,K: real,V: num] :
      ( ( ( restri3384469710633717624al_num @ M @ A3 @ K )
        = ( some_num @ V ) )
      = ( ( ( M @ K )
          = ( some_num @ V ) )
        & ( member_real @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5894_restrict__map__eq_I2_J,axiom,
    ! [M: int > option_num,A3: set_int,K: int,V: num] :
      ( ( ( restrict_map_int_num @ M @ A3 @ K )
        = ( some_num @ V ) )
      = ( ( ( M @ K )
          = ( some_num @ V ) )
        & ( member_int @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5895_restrict__map__eq_I2_J,axiom,
    ! [M: set_nat > option_nat,A3: set_set_nat,K: set_nat,V: nat] :
      ( ( ( restri1785250428630319752at_nat @ M @ A3 @ K )
        = ( some_nat @ V ) )
      = ( ( ( M @ K )
          = ( some_nat @ V ) )
        & ( member_set_nat @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5896_restrict__map__eq_I2_J,axiom,
    ! [M: set_nat > option_num,A3: set_set_nat,K: set_nat,V: num] :
      ( ( ( restri7565954251640874194at_num @ M @ A3 @ K )
        = ( some_num @ V ) )
      = ( ( ( M @ K )
          = ( some_num @ V ) )
        & ( member_set_nat @ K @ A3 ) ) ) ).

% restrict_map_eq(2)
thf(fact_5897_total__on__empty,axiom,
    ! [R3: set_Pr6218003697084177305l_real] : ( total_on_real @ bot_bot_set_real @ R3 ) ).

% total_on_empty
thf(fact_5898_total__on__empty,axiom,
    ! [R3: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R3 ) ).

% total_on_empty
thf(fact_5899_total__on__empty,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R3 ) ).

% total_on_empty
thf(fact_5900_total__on__empty,axiom,
    ! [R3: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R3 ) ).

% total_on_empty
thf(fact_5901_pigeonhole__infinite,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite_finite_nat @ ( image_VEBT_VEBT_nat @ F @ A3 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5902_pigeonhole__infinite,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_nat @ ( image_real_nat @ F @ A3 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A2: real] :
                      ( ( member_real @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5903_pigeonhole__infinite,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite_finite_int @ ( image_VEBT_VEBT_int @ F @ A3 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5904_pigeonhole__infinite,axiom,
    ! [A3: set_real,F: real > int] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_int @ ( image_real_int @ F @ A3 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A2: real] :
                      ( ( member_real @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5905_pigeonhole__infinite,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > complex] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite3207457112153483333omplex @ ( image_3793382806556112285omplex @ F @ A3 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5906_pigeonhole__infinite,axiom,
    ! [A3: set_real,F: real > complex] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite3207457112153483333omplex @ ( image_real_complex @ F @ A3 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A2: real] :
                      ( ( member_real @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5907_pigeonhole__infinite,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ~ ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite6017078050557962740nteger @ ( image_2092689629700589388nteger @ F @ A3 ) )
       => ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
            & ~ ( finite5795047828879050333T_VEBT
                @ ( collect_VEBT_VEBT
                  @ ^ [A2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5908_pigeonhole__infinite,axiom,
    ! [A3: set_real,F: real > code_integer] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite6017078050557962740nteger @ ( image_4958697645175560720nteger @ F @ A3 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A3 )
            & ~ ( finite_finite_real
                @ ( collect_real
                  @ ^ [A2: real] :
                      ( ( member_real @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5909_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_real @ ( image_nat_real @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5910_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_5911_image__Collect__subsetI,axiom,
    ! [P: complex > $o,F: complex > nat,B4: set_nat] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ ( collect_complex @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5912_image__Collect__subsetI,axiom,
    ! [P: complex > $o,F: complex > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_932796090930683071T_VEBT @ F @ ( collect_complex @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5913_image__Collect__subsetI,axiom,
    ! [P: complex > $o,F: complex > real,B4: set_real] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_complex_real @ F @ ( collect_complex @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5914_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5915_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_nat_VEBT_VEBT @ F @ ( collect_nat @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5916_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > real,B4: set_real] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_nat_real @ F @ ( collect_nat @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5917_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > nat,B4: set_nat] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( collect_int @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5918_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_VEBT_VEBT @ ( F @ X3 ) @ B4 ) )
     => ( ord_le4337996190870823476T_VEBT @ ( image_int_VEBT_VEBT @ F @ ( collect_int @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5919_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > real,B4: set_real] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_real @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_real @ ( image_int_real @ F @ ( collect_int @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5920_image__Collect__subsetI,axiom,
    ! [P: complex > $o,F: complex > int,B4: set_int] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( member_int @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_int @ ( image_complex_int @ F @ ( collect_complex @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_5921_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_5922_all__finite__subset__image,axiom,
    ! [F: nat > real,A3: set_nat,P: set_real > $o] :
      ( ( ! [B5: set_real] :
            ( ( ( finite_finite_real @ B5 )
              & ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A3 ) )
           => ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5923_all__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A3 ) )
           => ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5924_all__finite__subset__image,axiom,
    ! [F: complex > nat,A3: set_complex,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_complex_nat @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ A3 ) )
           => ( P @ ( image_complex_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5925_all__finite__subset__image,axiom,
    ! [F: code_integer > nat,A3: set_Code_integer,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_951025933927791156er_nat @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ A3 ) )
           => ( P @ ( image_951025933927791156er_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5926_all__finite__subset__image,axiom,
    ! [F: nat > complex,A3: set_nat,P: set_complex > $o] :
      ( ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ ( image_nat_complex @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A3 ) )
           => ( P @ ( image_nat_complex @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5927_all__finite__subset__image,axiom,
    ! [F: complex > complex,A3: set_complex,P: set_complex > $o] :
      ( ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ ( image_1468599708987790691omplex @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ A3 ) )
           => ( P @ ( image_1468599708987790691omplex @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5928_all__finite__subset__image,axiom,
    ! [F: code_integer > complex,A3: set_Code_integer,P: set_complex > $o] :
      ( ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ ( image_3397630267976458002omplex @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ A3 ) )
           => ( P @ ( image_3397630267976458002omplex @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5929_all__finite__subset__image,axiom,
    ! [F: nat > code_integer,A3: set_nat,P: set_Code_integer > $o] :
      ( ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ ( image_1215581382706833972nteger @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A3 ) )
           => ( P @ ( image_1215581382706833972nteger @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5930_all__finite__subset__image,axiom,
    ! [F: complex > code_integer,A3: set_complex,P: set_Code_integer > $o] :
      ( ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ ( image_1994230757181692690nteger @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_complex] :
            ( ( ( finite3207457112153483333omplex @ B5 )
              & ( ord_le211207098394363844omplex @ B5 @ A3 ) )
           => ( P @ ( image_1994230757181692690nteger @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5931_all__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ ( image_4470545334726330049nteger @ F @ A3 ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B5 )
              & ( ord_le7084787975880047091nteger @ B5 @ A3 ) )
           => ( P @ ( image_4470545334726330049nteger @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_5932_ex__finite__subset__image,axiom,
    ! [F: nat > real,A3: set_nat,P: set_real > $o] :
      ( ( ? [B5: set_real] :
            ( ( finite_finite_real @ B5 )
            & ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A3 )
            & ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5933_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A3 )
            & ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5934_ex__finite__subset__image,axiom,
    ! [F: complex > nat,A3: set_complex,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_complex_nat @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ A3 )
            & ( P @ ( image_complex_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5935_ex__finite__subset__image,axiom,
    ! [F: code_integer > nat,A3: set_Code_integer,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_951025933927791156er_nat @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ A3 )
            & ( P @ ( image_951025933927791156er_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5936_ex__finite__subset__image,axiom,
    ! [F: nat > complex,A3: set_nat,P: set_complex > $o] :
      ( ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ ( image_nat_complex @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A3 )
            & ( P @ ( image_nat_complex @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5937_ex__finite__subset__image,axiom,
    ! [F: complex > complex,A3: set_complex,P: set_complex > $o] :
      ( ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ ( image_1468599708987790691omplex @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ A3 )
            & ( P @ ( image_1468599708987790691omplex @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5938_ex__finite__subset__image,axiom,
    ! [F: code_integer > complex,A3: set_Code_integer,P: set_complex > $o] :
      ( ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ ( image_3397630267976458002omplex @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ A3 )
            & ( P @ ( image_3397630267976458002omplex @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5939_ex__finite__subset__image,axiom,
    ! [F: nat > code_integer,A3: set_nat,P: set_Code_integer > $o] :
      ( ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ ( image_1215581382706833972nteger @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A3 )
            & ( P @ ( image_1215581382706833972nteger @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5940_ex__finite__subset__image,axiom,
    ! [F: complex > code_integer,A3: set_complex,P: set_Code_integer > $o] :
      ( ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ ( image_1994230757181692690nteger @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_complex] :
            ( ( finite3207457112153483333omplex @ B5 )
            & ( ord_le211207098394363844omplex @ B5 @ A3 )
            & ( P @ ( image_1994230757181692690nteger @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5941_ex__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ ( image_4470545334726330049nteger @ F @ A3 ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B5 )
            & ( ord_le7084787975880047091nteger @ B5 @ A3 )
            & ( P @ ( image_4470545334726330049nteger @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_5942_finite__subset__image,axiom,
    ! [B4: set_real,F: nat > real,A3: set_nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A3 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
            & ( finite_finite_nat @ C6 )
            & ( B4
              = ( image_nat_real @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5943_finite__subset__image,axiom,
    ! [B4: set_nat,F: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
            & ( finite_finite_nat @ C6 )
            & ( B4
              = ( image_nat_nat @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5944_finite__subset__image,axiom,
    ! [B4: set_nat,F: complex > nat,A3: set_complex] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_complex_nat @ F @ A3 ) )
       => ? [C6: set_complex] :
            ( ( ord_le211207098394363844omplex @ C6 @ A3 )
            & ( finite3207457112153483333omplex @ C6 )
            & ( B4
              = ( image_complex_nat @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5945_finite__subset__image,axiom,
    ! [B4: set_nat,F: code_integer > nat,A3: set_Code_integer] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_951025933927791156er_nat @ F @ A3 ) )
       => ? [C6: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C6 @ A3 )
            & ( finite6017078050557962740nteger @ C6 )
            & ( B4
              = ( image_951025933927791156er_nat @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5946_finite__subset__image,axiom,
    ! [B4: set_complex,F: nat > complex,A3: set_nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A3 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
            & ( finite_finite_nat @ C6 )
            & ( B4
              = ( image_nat_complex @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5947_finite__subset__image,axiom,
    ! [B4: set_complex,F: complex > complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_1468599708987790691omplex @ F @ A3 ) )
       => ? [C6: set_complex] :
            ( ( ord_le211207098394363844omplex @ C6 @ A3 )
            & ( finite3207457112153483333omplex @ C6 )
            & ( B4
              = ( image_1468599708987790691omplex @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5948_finite__subset__image,axiom,
    ! [B4: set_complex,F: code_integer > complex,A3: set_Code_integer] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_3397630267976458002omplex @ F @ A3 ) )
       => ? [C6: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C6 @ A3 )
            & ( finite6017078050557962740nteger @ C6 )
            & ( B4
              = ( image_3397630267976458002omplex @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5949_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: nat > code_integer,A3: set_nat] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A3 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A3 )
            & ( finite_finite_nat @ C6 )
            & ( B4
              = ( image_1215581382706833972nteger @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5950_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: complex > code_integer,A3: set_complex] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1994230757181692690nteger @ F @ A3 ) )
       => ? [C6: set_complex] :
            ( ( ord_le211207098394363844omplex @ C6 @ A3 )
            & ( finite3207457112153483333omplex @ C6 )
            & ( B4
              = ( image_1994230757181692690nteger @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5951_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: code_integer > code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A3 ) )
       => ? [C6: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C6 @ A3 )
            & ( finite6017078050557962740nteger @ C6 )
            & ( B4
              = ( image_4470545334726330049nteger @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_5952_finite__surj,axiom,
    ! [A3: set_nat,B4: set_real,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_real @ B4 @ ( image_nat_real @ F @ A3 ) )
       => ( finite_finite_real @ B4 ) ) ) ).

% finite_surj
thf(fact_5953_finite__surj,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_5954_finite__surj,axiom,
    ! [A3: set_nat,B4: set_complex,F: nat > complex] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_nat_complex @ F @ A3 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_5955_finite__surj,axiom,
    ! [A3: set_nat,B4: set_Code_integer,F: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_5956_finite__surj,axiom,
    ! [A3: set_int,B4: set_nat,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_int_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_5957_finite__surj,axiom,
    ! [A3: set_int,B4: set_complex,F: int > complex] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_int_complex @ F @ A3 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_5958_finite__surj,axiom,
    ! [A3: set_int,B4: set_Code_integer,F: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1587234942943678608nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_5959_finite__surj,axiom,
    ! [A3: set_complex,B4: set_nat,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_complex_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_5960_finite__surj,axiom,
    ! [A3: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le211207098394363844omplex @ B4 @ ( image_1468599708987790691omplex @ F @ A3 ) )
       => ( finite3207457112153483333omplex @ B4 ) ) ) ).

% finite_surj
thf(fact_5961_finite__surj,axiom,
    ! [A3: set_complex,B4: set_Code_integer,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1994230757181692690nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B4 ) ) ) ).

% finite_surj
thf(fact_5962_image__diff__subset,axiom,
    ! [F: nat > real,A3: set_nat,B4: set_nat] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ ( image_nat_real @ F @ A3 ) @ ( image_nat_real @ F @ B4 ) ) @ ( image_nat_real @ F @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_5963_image__diff__subset,axiom,
    ! [F: nat > set_nat,A3: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_5964_image__diff__subset,axiom,
    ! [F: nat > nat,A3: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B4 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_5965_image__diff__subset,axiom,
    ! [F: int > int,A3: set_int,B4: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B4 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A3 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_5966_image__diff__subset,axiom,
    ! [F: nat > int,A3: set_nat,B4: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B4 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_5967_nat__seg__image__imp__finite,axiom,
    ! [A3: set_real,F: nat > real,N: nat] :
      ( ( A3
        = ( image_nat_real @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite_finite_real @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5968_nat__seg__image__imp__finite,axiom,
    ! [A3: set_set_nat,F: nat > set_nat,N: nat] :
      ( ( A3
        = ( image_nat_set_nat @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5969_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F: nat > nat,N: nat] :
      ( ( A3
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5970_nat__seg__image__imp__finite,axiom,
    ! [A3: set_int,F: nat > int,N: nat] :
      ( ( A3
        = ( image_nat_int @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite_finite_int @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5971_nat__seg__image__imp__finite,axiom,
    ! [A3: set_complex,F: nat > complex,N: nat] :
      ( ( A3
        = ( image_nat_complex @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite3207457112153483333omplex @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5972_nat__seg__image__imp__finite,axiom,
    ! [A3: set_Code_integer,F: nat > code_integer,N: nat] :
      ( ( A3
        = ( image_1215581382706833972nteger @ F
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
     => ( finite6017078050557962740nteger @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_5973_finite__conv__nat__seg__image,axiom,
    ( finite_finite_real
    = ( ^ [A5: set_real] :
        ? [N4: nat,F4: nat > real] :
          ( A5
          = ( image_nat_real @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5974_finite__conv__nat__seg__image,axiom,
    ( finite1152437895449049373et_nat
    = ( ^ [A5: set_set_nat] :
        ? [N4: nat,F4: nat > set_nat] :
          ( A5
          = ( image_nat_set_nat @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5975_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A5: set_nat] :
        ? [N4: nat,F4: nat > nat] :
          ( A5
          = ( image_nat_nat @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5976_finite__conv__nat__seg__image,axiom,
    ( finite_finite_int
    = ( ^ [A5: set_int] :
        ? [N4: nat,F4: nat > int] :
          ( A5
          = ( image_nat_int @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5977_finite__conv__nat__seg__image,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A5: set_complex] :
        ? [N4: nat,F4: nat > complex] :
          ( A5
          = ( image_nat_complex @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5978_finite__conv__nat__seg__image,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A5: set_Code_integer] :
        ? [N4: nat,F4: nat > code_integer] :
          ( A5
          = ( image_1215581382706833972nteger @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_5979_image__constant,axiom,
    ! [X4: nat,A3: set_nat,C: vEBT_VEBT] :
      ( ( member_nat @ X4 @ A3 )
     => ( ( image_nat_VEBT_VEBT
          @ ^ [X: nat] : C
          @ A3 )
        = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_5980_image__constant,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( image_3375948659692109573T_VEBT
          @ ^ [X: vEBT_VEBT] : C
          @ A3 )
        = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_5981_image__constant,axiom,
    ! [X4: real,A3: set_real,C: vEBT_VEBT] :
      ( ( member_real @ X4 @ A3 )
     => ( ( image_real_VEBT_VEBT
          @ ^ [X: real] : C
          @ A3 )
        = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_5982_image__constant,axiom,
    ! [X4: int,A3: set_int,C: vEBT_VEBT] :
      ( ( member_int @ X4 @ A3 )
     => ( ( image_int_VEBT_VEBT
          @ ^ [X: int] : C
          @ A3 )
        = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% image_constant
thf(fact_5983_image__constant,axiom,
    ! [X4: nat,A3: set_nat,C: real] :
      ( ( member_nat @ X4 @ A3 )
     => ( ( image_nat_real
          @ ^ [X: nat] : C
          @ A3 )
        = ( insert_real @ C @ bot_bot_set_real ) ) ) ).

% image_constant
thf(fact_5984_image__constant,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,C: real] :
      ( ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( image_VEBT_VEBT_real
          @ ^ [X: vEBT_VEBT] : C
          @ A3 )
        = ( insert_real @ C @ bot_bot_set_real ) ) ) ).

% image_constant
thf(fact_5985_image__constant,axiom,
    ! [X4: real,A3: set_real,C: real] :
      ( ( member_real @ X4 @ A3 )
     => ( ( image_real_real
          @ ^ [X: real] : C
          @ A3 )
        = ( insert_real @ C @ bot_bot_set_real ) ) ) ).

% image_constant
thf(fact_5986_image__constant,axiom,
    ! [X4: int,A3: set_int,C: real] :
      ( ( member_int @ X4 @ A3 )
     => ( ( image_int_real
          @ ^ [X: int] : C
          @ A3 )
        = ( insert_real @ C @ bot_bot_set_real ) ) ) ).

% image_constant
thf(fact_5987_image__constant,axiom,
    ! [X4: nat,A3: set_nat,C: $o] :
      ( ( member_nat @ X4 @ A3 )
     => ( ( image_nat_o
          @ ^ [X: nat] : C
          @ A3 )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_5988_image__constant,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,C: $o] :
      ( ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( image_VEBT_VEBT_o
          @ ^ [X: vEBT_VEBT] : C
          @ A3 )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_5989_image__constant__conv,axiom,
    ! [A3: set_real,C: vEBT_VEBT] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( image_real_VEBT_VEBT
            @ ^ [X: real] : C
            @ A3 )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( image_real_VEBT_VEBT
            @ ^ [X: real] : C
            @ A3 )
          = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% image_constant_conv
thf(fact_5990_image__constant__conv,axiom,
    ! [A3: set_real,C: real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( image_real_real
            @ ^ [X: real] : C
            @ A3 )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( image_real_real
            @ ^ [X: real] : C
            @ A3 )
          = ( insert_real @ C @ bot_bot_set_real ) ) ) ) ).

% image_constant_conv
thf(fact_5991_image__constant__conv,axiom,
    ! [A3: set_real,C: $o] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( image_real_o
            @ ^ [X: real] : C
            @ A3 )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( image_real_o
            @ ^ [X: real] : C
            @ A3 )
          = ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_5992_image__constant__conv,axiom,
    ! [A3: set_real,C: nat] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( image_real_nat
            @ ^ [X: real] : C
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( image_real_nat
            @ ^ [X: real] : C
            @ A3 )
          = ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_5993_image__constant__conv,axiom,
    ! [A3: set_real,C: int] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( image_real_int
            @ ^ [X: real] : C
            @ A3 )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( image_real_int
            @ ^ [X: real] : C
            @ A3 )
          = ( insert_int @ C @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_5994_image__constant__conv,axiom,
    ! [A3: set_o,C: vEBT_VEBT] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_VEBT_VEBT
            @ ^ [X: $o] : C
            @ A3 )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_VEBT_VEBT
            @ ^ [X: $o] : C
            @ A3 )
          = ( insert_VEBT_VEBT @ C @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% image_constant_conv
thf(fact_5995_image__constant__conv,axiom,
    ! [A3: set_o,C: real] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_real
            @ ^ [X: $o] : C
            @ A3 )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_real
            @ ^ [X: $o] : C
            @ A3 )
          = ( insert_real @ C @ bot_bot_set_real ) ) ) ) ).

% image_constant_conv
thf(fact_5996_image__constant__conv,axiom,
    ! [A3: set_o,C: $o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X: $o] : C
            @ A3 )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X: $o] : C
            @ A3 )
          = ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_5997_image__constant__conv,axiom,
    ! [A3: set_o,C: nat] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X: $o] : C
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X: $o] : C
            @ A3 )
          = ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_5998_image__constant__conv,axiom,
    ! [A3: set_o,C: int] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_int
            @ ^ [X: $o] : C
            @ A3 )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_int
            @ ^ [X: $o] : C
            @ A3 )
          = ( insert_int @ C @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_5999_zip__same__conv__map,axiom,
    ! [Xs2: list_o] :
      ( ( zip_o_o @ Xs2 @ Xs2 )
      = ( map_o_3702434973371374163od_o_o
        @ ^ [X: $o] : ( product_Pair_o_o @ X @ X )
        @ Xs2 ) ) ).

% zip_same_conv_map
thf(fact_6000_zip__same__conv__map,axiom,
    ! [Xs2: list_nat] :
      ( ( zip_nat_nat @ Xs2 @ Xs2 )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ Xs2 ) ) ).

% zip_same_conv_map
thf(fact_6001_zip__same__conv__map,axiom,
    ! [Xs2: list_int] :
      ( ( zip_int_int @ Xs2 @ Xs2 )
      = ( map_in7157766398909135175nt_int
        @ ^ [X: int] : ( product_Pair_int_int @ X @ X )
        @ Xs2 ) ) ).

% zip_same_conv_map
thf(fact_6002_zip__same__conv__map,axiom,
    ! [Xs2: list_Code_integer] :
      ( ( zip_Co3543743374963494515nteger @ Xs2 @ Xs2 )
      = ( map_Co3589949550033412536nteger
        @ ^ [X: code_integer] : ( produc1086072967326762835nteger @ X @ X )
        @ Xs2 ) ) ).

% zip_same_conv_map
thf(fact_6003_pair__list__split,axiom,
    ! [L: list_P8689742595348180415l_real] :
      ~ ! [L12: list_real,L23: list_real] :
          ( ( L
            = ( zip_real_real @ L12 @ L23 ) )
         => ( ( ( size_size_list_real @ L12 )
              = ( size_size_list_real @ L23 ) )
           => ( ( size_s3932428310213730859l_real @ L )
             != ( size_size_list_real @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6004_pair__list__split,axiom,
    ! [L: list_P3595434254542482545real_o] :
      ~ ! [L12: list_real,L23: list_o] :
          ( ( L
            = ( zip_real_o @ L12 @ L23 ) )
         => ( ( ( size_size_list_real @ L12 )
              = ( size_size_list_o @ L23 ) )
           => ( ( size_s987546567493390085real_o @ L )
             != ( size_size_list_o @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6005_pair__list__split,axiom,
    ! [L: list_P6834414599653733731al_nat] :
      ~ ! [L12: list_real,L23: list_nat] :
          ( ( L
            = ( zip_real_nat @ L12 @ L23 ) )
         => ( ( ( size_size_list_real @ L12 )
              = ( size_size_list_nat @ L23 ) )
           => ( ( size_s1877336372972134351al_nat @ L )
             != ( size_size_list_nat @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6006_pair__list__split,axiom,
    ! [L: list_P4344331454722006975al_int] :
      ~ ! [L12: list_real,L23: list_int] :
          ( ( L
            = ( zip_real_int @ L12 @ L23 ) )
         => ( ( ( size_size_list_real @ L12 )
              = ( size_size_list_int @ L23 ) )
           => ( ( size_s8610625264895183403al_int @ L )
             != ( size_size_list_int @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6007_pair__list__split,axiom,
    ! [L: list_P5232166724548748803o_real] :
      ~ ! [L12: list_o,L23: list_real] :
          ( ( L
            = ( zip_o_real @ L12 @ L23 ) )
         => ( ( ( size_size_list_o @ L12 )
              = ( size_size_list_real @ L23 ) )
           => ( ( size_s2624279037499656343o_real @ L )
             != ( size_size_list_real @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6008_pair__list__split,axiom,
    ! [L: list_P4002435161011370285od_o_o] :
      ~ ! [L12: list_o,L23: list_o] :
          ( ( L
            = ( zip_o_o @ L12 @ L23 ) )
         => ( ( ( size_size_list_o @ L12 )
              = ( size_size_list_o @ L23 ) )
           => ( ( size_s1515746228057227161od_o_o @ L )
             != ( size_size_list_o @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6009_pair__list__split,axiom,
    ! [L: list_P6285523579766656935_o_nat] :
      ~ ! [L12: list_o,L23: list_nat] :
          ( ( L
            = ( zip_o_nat @ L12 @ L23 ) )
         => ( ( ( size_size_list_o @ L12 )
              = ( size_size_list_nat @ L23 ) )
           => ( ( size_s5443766701097040955_o_nat @ L )
             != ( size_size_list_nat @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6010_pair__list__split,axiom,
    ! [L: list_P3795440434834930179_o_int] :
      ~ ! [L12: list_o,L23: list_int] :
          ( ( L
            = ( zip_o_int @ L12 @ L23 ) )
         => ( ( ( size_size_list_o @ L12 )
              = ( size_size_list_int @ L23 ) )
           => ( ( size_s2953683556165314199_o_int @ L )
             != ( size_size_list_int @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6011_pair__list__split,axiom,
    ! [L: list_P3644420460460130531t_real] :
      ~ ! [L12: list_nat,L23: list_real] :
          ( ( L
            = ( zip_nat_real @ L12 @ L23 ) )
         => ( ( ( size_size_list_nat @ L12 )
              = ( size_size_list_real @ L23 ) )
           => ( ( size_s7910714270633306959t_real @ L )
             != ( size_size_list_real @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6012_pair__list__split,axiom,
    ! [L: list_P7333126701944960589_nat_o] :
      ~ ! [L12: list_nat,L23: list_o] :
          ( ( L
            = ( zip_nat_o @ L12 @ L23 ) )
         => ( ( ( size_size_list_nat @ L12 )
              = ( size_size_list_o @ L23 ) )
           => ( ( size_s6491369823275344609_nat_o @ L )
             != ( size_size_list_o @ L23 ) ) ) ) ).

% pair_list_split
thf(fact_6013_the__elem__image__unique,axiom,
    ! [A3: set_nat,F: nat > real,X4: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A3 )
           => ( ( F @ Y3 )
              = ( F @ X4 ) ) )
       => ( ( the_elem_real @ ( image_nat_real @ F @ A3 ) )
          = ( F @ X4 ) ) ) ) ).

% the_elem_image_unique
thf(fact_6014_the__elem__image__unique,axiom,
    ! [A3: set_nat,F: nat > set_nat,X4: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A3 )
           => ( ( F @ Y3 )
              = ( F @ X4 ) ) )
       => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = ( F @ X4 ) ) ) ) ).

% the_elem_image_unique
thf(fact_6015_the__elem__image__unique,axiom,
    ! [A3: set_nat,F: nat > nat,X4: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A3 )
           => ( ( F @ Y3 )
              = ( F @ X4 ) ) )
       => ( ( the_elem_nat @ ( image_nat_nat @ F @ A3 ) )
          = ( F @ X4 ) ) ) ) ).

% the_elem_image_unique
thf(fact_6016_the__elem__image__unique,axiom,
    ! [A3: set_nat,F: nat > int,X4: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A3 )
           => ( ( F @ Y3 )
              = ( F @ X4 ) ) )
       => ( ( the_elem_int @ ( image_nat_int @ F @ A3 ) )
          = ( F @ X4 ) ) ) ) ).

% the_elem_image_unique
thf(fact_6017_the__elem__image__unique,axiom,
    ! [A3: set_int,F: int > int,X4: int] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A3 )
           => ( ( F @ Y3 )
              = ( F @ X4 ) ) )
       => ( ( the_elem_int @ ( image_int_int @ F @ A3 ) )
          = ( F @ X4 ) ) ) ) ).

% the_elem_image_unique
thf(fact_6018_restrict__map__upd,axiom,
    ! [F: nat > option_nat,S3: set_nat,K: nat,V: nat] :
      ( ( fun_up1493157387958331631on_nat @ ( restrict_map_nat_nat @ F @ S3 ) @ K @ ( some_nat @ V ) )
      = ( restrict_map_nat_nat @ ( fun_up1493157387958331631on_nat @ F @ K @ ( some_nat @ V ) ) @ ( insert_nat @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6019_restrict__map__upd,axiom,
    ! [F: vEBT_VEBT > option_nat,S3: set_VEBT_VEBT,K: vEBT_VEBT,V: nat] :
      ( ( fun_up5885881570350532375on_nat @ ( restri774867724463461460BT_nat @ F @ S3 ) @ K @ ( some_nat @ V ) )
      = ( restri774867724463461460BT_nat @ ( fun_up5885881570350532375on_nat @ F @ K @ ( some_nat @ V ) ) @ ( insert_VEBT_VEBT @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6020_restrict__map__upd,axiom,
    ! [F: int > option_nat,S3: set_int,K: int,V: nat] :
      ( ( fun_up3620524117960394059on_nat @ ( restrict_map_int_nat @ F @ S3 ) @ K @ ( some_nat @ V ) )
      = ( restrict_map_int_nat @ ( fun_up3620524117960394059on_nat @ F @ K @ ( some_nat @ V ) ) @ ( insert_int @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6021_restrict__map__upd,axiom,
    ! [F: $o > option_nat,S3: set_o,K: $o,V: nat] :
      ( ( fun_upd_o_option_nat @ ( restrict_map_o_nat @ F @ S3 ) @ K @ ( some_nat @ V ) )
      = ( restrict_map_o_nat @ ( fun_upd_o_option_nat @ F @ K @ ( some_nat @ V ) ) @ ( insert_o @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6022_restrict__map__upd,axiom,
    ! [F: real > option_nat,S3: set_real,K: real,V: nat] :
      ( ( fun_up6677080212936659659on_nat @ ( restri6827137924477938990al_nat @ F @ S3 ) @ K @ ( some_nat @ V ) )
      = ( restri6827137924477938990al_nat @ ( fun_up6677080212936659659on_nat @ F @ K @ ( some_nat @ V ) ) @ ( insert_real @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6023_restrict__map__upd,axiom,
    ! [F: nat > option_num,S3: set_nat,K: nat,V: num] :
      ( ( fun_up2201401324907169337on_num @ ( restrict_map_nat_num @ F @ S3 ) @ K @ ( some_num @ V ) )
      = ( restrict_map_nat_num @ ( fun_up2201401324907169337on_num @ F @ K @ ( some_num @ V ) ) @ ( insert_nat @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6024_restrict__map__upd,axiom,
    ! [F: vEBT_VEBT > option_num,S3: set_VEBT_VEBT,K: vEBT_VEBT,V: num] :
      ( ( fun_up6594125507299370081on_num @ ( restri6555571547474015902BT_num @ F @ S3 ) @ K @ ( some_num @ V ) )
      = ( restri6555571547474015902BT_num @ ( fun_up6594125507299370081on_num @ F @ K @ ( some_num @ V ) ) @ ( insert_VEBT_VEBT @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6025_restrict__map__upd,axiom,
    ! [F: int > option_num,S3: set_int,K: int,V: num] :
      ( ( fun_up4328768054909231765on_num @ ( restrict_map_int_num @ F @ S3 ) @ K @ ( some_num @ V ) )
      = ( restrict_map_int_num @ ( fun_up4328768054909231765on_num @ F @ K @ ( some_num @ V ) ) @ ( insert_int @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6026_restrict__map__upd,axiom,
    ! [F: $o > option_num,S3: set_o,K: $o,V: num] :
      ( ( fun_upd_o_option_num @ ( restrict_map_o_num @ F @ S3 ) @ K @ ( some_num @ V ) )
      = ( restrict_map_o_num @ ( fun_upd_o_option_num @ F @ K @ ( some_num @ V ) ) @ ( insert_o @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6027_restrict__map__upd,axiom,
    ! [F: real > option_num,S3: set_real,K: real,V: num] :
      ( ( fun_up7385324149885497365on_num @ ( restri3384469710633717624al_num @ F @ S3 ) @ K @ ( some_num @ V ) )
      = ( restri3384469710633717624al_num @ ( fun_up7385324149885497365on_num @ F @ K @ ( some_num @ V ) ) @ ( insert_real @ K @ S3 ) ) ) ).

% restrict_map_upd
thf(fact_6028_graph__restrictD_I2_J,axiom,
    ! [K: int,V: int,M: int > option_int,A3: set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K @ V ) @ ( graph_int_int @ ( restrict_map_int_int @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_int @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_6029_graph__restrictD_I2_J,axiom,
    ! [K: code_integer,V: code_integer,M: code_integer > option_Code_integer,A3: set_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ K @ V ) @ ( graph_5282091004195177018nteger @ ( restri7010877615957751788nteger @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_Code_integer @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_6030_graph__restrictD_I2_J,axiom,
    ! [K: nat,V: nat,M: nat > option_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_6031_graph__restrictD_I2_J,axiom,
    ! [K: vEBT_VEBT,V: nat,M: vEBT_VEBT > option_nat,A3: set_VEBT_VEBT] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ K @ V ) @ ( graph_VEBT_VEBT_nat @ ( restri774867724463461460BT_nat @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_6032_graph__restrictD_I2_J,axiom,
    ! [K: nat,V: num,M: nat > option_num,A3: set_nat] :
      ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ K @ V ) @ ( graph_nat_num @ ( restrict_map_nat_num @ M @ A3 ) ) )
     => ( ( M @ K )
        = ( some_num @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_6033_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_6034_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_6035_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,Y: code_integer] :
      ( ( ( size_s3445333598471063425nteger @ Xs2 )
        = ( size_s3445333598471063425nteger @ Ys ) )
     => ( ( member_Code_integer @ Y @ ( set_Code_integer2 @ Ys ) )
       => ~ ! [X3: code_integer] :
              ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6036_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( 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 @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6037_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( 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 @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6038_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_real,Ys: list_real,Y: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( 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 @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6039_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_real,Ys: list_o,Y: $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X3: real] :
              ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X3 @ Y ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6040_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_real,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: real] :
              ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6041_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_real,Ys: list_int,Y: int] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( 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 @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6042_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X3 @ Y ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6043_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_real,Y: real] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X3 @ Y ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6044_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_o,Y: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_6045_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,X4: code_integer] :
      ( ( ( size_s3445333598471063425nteger @ Xs2 )
        = ( size_s3445333598471063425nteger @ Ys ) )
     => ( ( member_Code_integer @ X4 @ ( set_Code_integer2 @ Xs2 ) )
       => ~ ! [Y3: code_integer] :
              ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Y3 ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6046_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ~ ! [Y3: real] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6047_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ~ ! [Y3: $o] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6048_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6049_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ~ ! [Y3: int] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6050_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_real,X4: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: real] :
              ~ ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X4 @ Y3 ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6051_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_o,X4: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: $o] :
              ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X4 @ Y3 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6052_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_nat,X4: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X4 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6053_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_int,X4: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: int] :
              ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X4 @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6054_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_o,Ys: list_real,X4: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
       => ~ ! [Y3: real] :
              ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y3 ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_6055_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: nat > real] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L4 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L4 @ I2 ) ) ) )
       => ( ( image_nat_real @ F @ ( set_nat2 @ L ) )
          = ( image_nat_real @ F @ ( set_nat2 @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6056_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: nat > set_nat] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L4 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L4 @ I2 ) ) ) )
       => ( ( image_nat_set_nat @ F @ ( set_nat2 @ L ) )
          = ( image_nat_set_nat @ F @ ( set_nat2 @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6057_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L4 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L4 @ I2 ) ) ) )
       => ( ( image_nat_nat @ F @ ( set_nat2 @ L ) )
          = ( image_nat_nat @ F @ ( set_nat2 @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6058_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L4: list_nat,F: nat > int] :
      ( ( ( size_size_list_nat @ L )
        = ( size_size_list_nat @ L4 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( ( F @ ( nth_nat @ L @ I2 ) )
              = ( F @ ( nth_nat @ L4 @ I2 ) ) ) )
       => ( ( image_nat_int @ F @ ( set_nat2 @ L ) )
          = ( image_nat_int @ F @ ( set_nat2 @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6059_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L4: list_int,F: int > int] :
      ( ( ( size_size_list_int @ L )
        = ( size_size_list_int @ L4 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ L ) )
           => ( ( F @ ( nth_int @ L @ I2 ) )
              = ( F @ ( nth_int @ L4 @ I2 ) ) ) )
       => ( ( image_int_int @ F @ ( set_int2 @ L ) )
          = ( image_int_int @ F @ ( set_int2 @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6060_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > nat,X4: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_nat @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6061_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > vEBT_VEBT,X4: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBT @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6062_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > real,X4: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_real @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6063_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBTi > int,X4: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( ( member_int @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6064_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > nat,X4: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_nat @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6065_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,X4: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ ( F @ X4 ) @ ( 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 @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6066_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > real,X4: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_real @ ( F @ X4 ) @ ( image_VEBT_VEBT_real @ F @ ( set_VEBT_VEBT2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
            & ( ( F @ ( nth_VEBT_VEBT @ L @ I3 ) )
              = ( F @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6067_in__set__image__conv__nth,axiom,
    ! [F: vEBT_VEBT > int,X4: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( ( member_int @ ( F @ X4 ) @ ( image_VEBT_VEBT_int @ F @ ( set_VEBT_VEBT2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
            & ( ( F @ ( nth_VEBT_VEBT @ L @ I3 ) )
              = ( F @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6068_in__set__image__conv__nth,axiom,
    ! [F: real > nat,X4: real,L: list_real] :
      ( ( member_nat @ ( F @ X4 ) @ ( image_real_nat @ F @ ( set_real2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) )
            & ( ( F @ ( nth_real @ L @ I3 ) )
              = ( F @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6069_in__set__image__conv__nth,axiom,
    ! [F: real > vEBT_VEBT,X4: real,L: list_real] :
      ( ( member_VEBT_VEBT @ ( F @ X4 ) @ ( image_real_VEBT_VEBT @ F @ ( set_real2 @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) )
            & ( ( F @ ( nth_real @ L @ I3 ) )
              = ( F @ X4 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6070_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_6071_map__zip1,axiom,
    ! [K: nat,L: list_VEBT_VEBT] :
      ( ( map_VE6666121349297532579BT_nat
        @ ^ [X: vEBT_VEBT] : ( produc738532404422230701BT_nat @ X @ K )
        @ L )
      = ( zip_VEBT_VEBT_nat @ L @ ( replicate_nat @ ( size_s6755466524823107622T_VEBT @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6072_map__zip1,axiom,
    ! [K: code_integer,L: list_Code_integer] :
      ( ( map_Co3589949550033412536nteger
        @ ^ [X: code_integer] : ( produc1086072967326762835nteger @ X @ K )
        @ L )
      = ( zip_Co3543743374963494515nteger @ L @ ( replic7707675349574490269nteger @ ( size_s3445333598471063425nteger @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6073_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_real] :
      ( ( map_re7205069664741861231T_VEBT
        @ ^ [X: real] : ( produc6931449550656315951T_VEBT @ X @ K )
        @ L )
      = ( zip_real_VEBT_VEBT @ L @ ( replicate_VEBT_VEBT @ ( size_size_list_real @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6074_map__zip1,axiom,
    ! [K: $o,L: list_real] :
      ( ( map_re731699941522841299real_o
        @ ^ [X: real] : ( product_Pair_real_o @ X @ K )
        @ L )
      = ( zip_real_o @ L @ ( replicate_o @ ( size_size_list_real @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6075_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_o] :
      ( ( map_o_8925299737569714927T_VEBT
        @ ^ [X: $o] : ( produc2982872950893828659T_VEBT @ X @ K )
        @ L )
      = ( zip_o_VEBT_VEBT @ L @ ( replicate_VEBT_VEBT @ ( size_size_list_o @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6076_map__zip1,axiom,
    ! [K: $o,L: list_o] :
      ( ( map_o_3702434973371374163od_o_o
        @ ^ [X: $o] : ( product_Pair_o_o @ X @ K )
        @ L )
      = ( zip_o_o @ L @ ( replicate_o @ ( size_size_list_o @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6077_map__zip1,axiom,
    ! [K: vEBT_VEBT,L: list_nat] :
      ( ( map_na3584885621601055599T_VEBT
        @ ^ [X: nat] : ( produc599794634098209291T_VEBT @ X @ K )
        @ L )
      = ( zip_nat_VEBT_VEBT @ L @ ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6078_map__zip1,axiom,
    ! [K: $o,L: list_nat] :
      ( ( map_na6716429308333697747_nat_o
        @ ^ [X: nat] : ( product_Pair_nat_o @ X @ K )
        @ L )
      = ( zip_nat_o @ L @ ( replicate_o @ ( size_size_list_nat @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6079_map__zip1,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ K )
        @ L )
      = ( zip_nat_nat @ L @ ( replicate_nat @ ( size_size_list_nat @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6080_map__zip1,axiom,
    ! [K: num,L: list_nat] :
      ( ( map_na8006665559001981237at_num
        @ ^ [X: nat] : ( product_Pair_nat_num @ X @ K )
        @ L )
      = ( zip_nat_num @ L @ ( replicate_num @ ( size_size_list_nat @ L ) @ K ) ) ) ).

% map_zip1
thf(fact_6081_map__zip2,axiom,
    ! [K: code_integer,L: list_Code_integer] :
      ( ( map_Co3589949550033412536nteger @ ( produc1086072967326762835nteger @ K ) @ L )
      = ( zip_Co3543743374963494515nteger @ ( replic7707675349574490269nteger @ ( size_s3445333598471063425nteger @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6082_map__zip2,axiom,
    ! [K: nat,L: list_num] :
      ( ( map_nu4721551698833171051at_num @ ( product_Pair_nat_num @ K ) @ L )
      = ( zip_nat_num @ ( replicate_nat @ ( size_size_list_num @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6083_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_real] :
      ( ( map_re8618229306769252225T_real @ ( produc8117437818029410057T_real @ K ) @ L )
      = ( zip_VEBT_VEBT_real @ ( replicate_VEBT_VEBT @ ( size_size_list_real @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6084_map__zip2,axiom,
    ! [K: $o,L: list_real] :
      ( ( map_re7359128668140142949o_real @ ( product_Pair_o_real @ K ) @ L )
      = ( zip_o_real @ ( replicate_o @ ( size_size_list_real @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6085_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_o] :
      ( ( map_o_6754667662019005495VEBT_o @ ( produc8721562602347293563VEBT_o @ K ) @ L )
      = ( zip_VEBT_VEBT_o @ ( replicate_VEBT_VEBT @ ( size_size_list_o @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6086_map__zip2,axiom,
    ! [K: $o,L: list_o] :
      ( ( map_o_3702434973371374163od_o_o @ ( product_Pair_o_o @ K ) @ L )
      = ( zip_o_o @ ( replicate_o @ ( size_size_list_o @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6087_map__zip2,axiom,
    ! [K: $o,L: list_nat] :
      ( ( map_na3207894784278204717_o_nat @ ( product_Pair_o_nat @ K ) @ L )
      = ( zip_o_nat @ ( replicate_o @ ( size_size_list_nat @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6088_map__zip2,axiom,
    ! [K: nat,L: list_nat] :
      ( ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ K ) @ L )
      = ( zip_nat_nat @ ( replicate_nat @ ( size_size_list_nat @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6089_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_nat] :
      ( ( map_na4631810538828370761BT_nat @ ( produc738532404422230701BT_nat @ K ) @ L )
      = ( zip_VEBT_VEBT_nat @ ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6090_map__zip2,axiom,
    ! [K: vEBT_VEBT,L: list_int] :
      ( ( map_in8151279748432256513BT_int @ ( produc736041933913180425BT_int @ K ) @ L )
      = ( zip_VEBT_VEBT_int @ ( replicate_VEBT_VEBT @ ( size_size_list_int @ L ) @ K ) @ L ) ) ).

% map_zip2
thf(fact_6091_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > vEBT_VEBT] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_VEBT_VEBT @ F @ A3 )
        = ( finite7237651155313779962T_VEBT
          @ ^ [K3: nat] : ( insert_VEBT_VEBT @ ( F @ K3 ) )
          @ bot_bo8194388402131092736T_VEBT
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6092_image__fold__insert,axiom,
    ! [A3: set_int,F: int > vEBT_VEBT] :
      ( ( finite_finite_int @ A3 )
     => ( ( image_int_VEBT_VEBT @ F @ A3 )
        = ( finite7046945240081515934T_VEBT
          @ ^ [K3: int] : ( insert_VEBT_VEBT @ ( F @ K3 ) )
          @ bot_bo8194388402131092736T_VEBT
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6093_image__fold__insert,axiom,
    ! [A3: set_complex,F: complex > vEBT_VEBT] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( image_932796090930683071T_VEBT @ F @ A3 )
        = ( finite3784847964138203548T_VEBT
          @ ^ [K3: complex] : ( insert_VEBT_VEBT @ ( F @ K3 ) )
          @ bot_bo8194388402131092736T_VEBT
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6094_image__fold__insert,axiom,
    ! [A3: set_Code_integer,F: code_integer > vEBT_VEBT] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( image_2664455355285265616T_VEBT @ F @ A3 )
        = ( finite2306045077591485357T_VEBT
          @ ^ [K3: code_integer] : ( insert_VEBT_VEBT @ ( F @ K3 ) )
          @ bot_bo8194388402131092736T_VEBT
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6095_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_real @ F @ A3 )
        = ( finite6086821452261492220t_real
          @ ^ [K3: nat] : ( insert_real @ ( F @ K3 ) )
          @ bot_bot_set_real
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6096_image__fold__insert,axiom,
    ! [A3: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( image_int_real @ F @ A3 )
        = ( finite2872259066402291288t_real
          @ ^ [K3: int] : ( insert_real @ ( F @ K3 ) )
          @ bot_bot_set_real
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6097_image__fold__insert,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( image_complex_real @ F @ A3 )
        = ( finite2789990275036127706t_real
          @ ^ [K3: complex] : ( insert_real @ ( F @ K3 ) )
          @ bot_bot_set_real
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6098_image__fold__insert,axiom,
    ! [A3: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( image_7738145705984076560r_real @ F @ A3 )
        = ( finite7676789782520586249t_real
          @ ^ [K3: code_integer] : ( insert_real @ ( F @ K3 ) )
          @ bot_bot_set_real
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6099_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_o @ F @ A3 )
        = ( finite3217087857726763998_set_o
          @ ^ [K3: nat] : ( insert_o @ ( F @ K3 ) )
          @ bot_bot_set_o
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6100_image__fold__insert,axiom,
    ! [A3: set_int,F: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( image_int_o @ F @ A3 )
        = ( finite3903663890767314562_set_o
          @ ^ [K3: int] : ( insert_o @ ( F @ K3 ) )
          @ bot_bot_set_o
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_6101_fun__upd__image,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > nat,Y: nat] :
      ( ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_nat @ ( fun_up6512855943550542919BT_nat @ F @ X4 @ Y ) @ A3 )
          = ( insert_nat @ Y @ ( image_VEBT_VEBT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_nat @ ( fun_up6512855943550542919BT_nat @ F @ X4 @ Y ) @ A3 )
          = ( image_VEBT_VEBT_nat @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6102_fun__upd__image,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_3375948659692109573T_VEBT @ ( fun_up224749957652071293T_VEBT @ F @ X4 @ Y ) @ A3 )
          = ( insert_VEBT_VEBT @ Y @ ( image_3375948659692109573T_VEBT @ F @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_3375948659692109573T_VEBT @ ( fun_up224749957652071293T_VEBT @ F @ X4 @ Y ) @ A3 )
          = ( image_3375948659692109573T_VEBT @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6103_fun__upd__image,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > int,Y: int] :
      ( ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_int @ ( fun_up6510365473041492643BT_int @ F @ X4 @ Y ) @ A3 )
          = ( insert_int @ Y @ ( image_VEBT_VEBT_int @ F @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_int @ ( fun_up6510365473041492643BT_int @ F @ X4 @ Y ) @ A3 )
          = ( image_VEBT_VEBT_int @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6104_fun__upd__image,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > $o,Y: $o] :
      ( ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_o @ ( fun_upd_VEBT_VEBT_o @ F @ X4 @ Y ) @ A3 )
          = ( insert_o @ Y @ ( image_VEBT_VEBT_o @ F @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_o @ ( fun_upd_VEBT_VEBT_o @ F @ X4 @ Y ) @ A3 )
          = ( image_VEBT_VEBT_o @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6105_fun__upd__image,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > real,Y: real] :
      ( ( ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_real @ ( fun_up7749720967766031267T_real @ F @ X4 @ Y ) @ A3 )
          = ( insert_real @ Y @ ( image_VEBT_VEBT_real @ F @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
       => ( ( image_VEBT_VEBT_real @ ( fun_up7749720967766031267T_real @ F @ X4 @ Y ) @ A3 )
          = ( image_VEBT_VEBT_real @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6106_fun__upd__image,axiom,
    ! [X4: real,A3: set_real,F: real > nat,Y: nat] :
      ( ( ( member_real @ X4 @ A3 )
       => ( ( image_real_nat @ ( fun_upd_real_nat @ F @ X4 @ Y ) @ A3 )
          = ( insert_nat @ Y @ ( image_real_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
      & ( ~ ( member_real @ X4 @ A3 )
       => ( ( image_real_nat @ ( fun_upd_real_nat @ F @ X4 @ Y ) @ A3 )
          = ( image_real_nat @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6107_fun__upd__image,axiom,
    ! [X4: real,A3: set_real,F: real > vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( member_real @ X4 @ A3 )
       => ( ( image_real_VEBT_VEBT @ ( fun_up6563732700392937161T_VEBT @ F @ X4 @ Y ) @ A3 )
          = ( insert_VEBT_VEBT @ Y @ ( image_real_VEBT_VEBT @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
      & ( ~ ( member_real @ X4 @ A3 )
       => ( ( image_real_VEBT_VEBT @ ( fun_up6563732700392937161T_VEBT @ F @ X4 @ Y ) @ A3 )
          = ( image_real_VEBT_VEBT @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6108_fun__upd__image,axiom,
    ! [X4: real,A3: set_real,F: real > int,Y: int] :
      ( ( ( member_real @ X4 @ A3 )
       => ( ( image_real_int @ ( fun_upd_real_int @ F @ X4 @ Y ) @ A3 )
          = ( insert_int @ Y @ ( image_real_int @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
      & ( ~ ( member_real @ X4 @ A3 )
       => ( ( image_real_int @ ( fun_upd_real_int @ F @ X4 @ Y ) @ A3 )
          = ( image_real_int @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6109_fun__upd__image,axiom,
    ! [X4: real,A3: set_real,F: real > $o,Y: $o] :
      ( ( ( member_real @ X4 @ A3 )
       => ( ( image_real_o @ ( fun_upd_real_o @ F @ X4 @ Y ) @ A3 )
          = ( insert_o @ Y @ ( image_real_o @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
      & ( ~ ( member_real @ X4 @ A3 )
       => ( ( image_real_o @ ( fun_upd_real_o @ F @ X4 @ Y ) @ A3 )
          = ( image_real_o @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6110_fun__upd__image,axiom,
    ! [X4: real,A3: set_real,F: real > real,Y: real] :
      ( ( ( member_real @ X4 @ A3 )
       => ( ( image_real_real @ ( fun_upd_real_real @ F @ X4 @ Y ) @ A3 )
          = ( insert_real @ Y @ ( image_real_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
      & ( ~ ( member_real @ X4 @ A3 )
       => ( ( image_real_real @ ( fun_upd_real_real @ F @ X4 @ Y ) @ A3 )
          = ( image_real_real @ F @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_6111_translation__subtract__diff,axiom,
    ! [A: real,S2: set_real,T: set_real] :
      ( ( image_real_real
        @ ^ [X: real] : ( minus_minus_real @ X @ A )
        @ ( minus_minus_set_real @ S2 @ T ) )
      = ( minus_minus_set_real
        @ ( image_real_real
          @ ^ [X: real] : ( minus_minus_real @ X @ A )
          @ S2 )
        @ ( image_real_real
          @ ^ [X: real] : ( minus_minus_real @ X @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_6112_translation__subtract__diff,axiom,
    ! [A: rat,S2: set_rat,T: set_rat] :
      ( ( image_rat_rat
        @ ^ [X: rat] : ( minus_minus_rat @ X @ A )
        @ ( minus_minus_set_rat @ S2 @ T ) )
      = ( minus_minus_set_rat
        @ ( image_rat_rat
          @ ^ [X: rat] : ( minus_minus_rat @ X @ A )
          @ S2 )
        @ ( image_rat_rat
          @ ^ [X: rat] : ( minus_minus_rat @ X @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_6113_translation__subtract__diff,axiom,
    ! [A: int,S2: set_int,T: set_int] :
      ( ( image_int_int
        @ ^ [X: int] : ( minus_minus_int @ X @ A )
        @ ( minus_minus_set_int @ S2 @ T ) )
      = ( minus_minus_set_int
        @ ( image_int_int
          @ ^ [X: int] : ( minus_minus_int @ X @ A )
          @ S2 )
        @ ( image_int_int
          @ ^ [X: int] : ( minus_minus_int @ X @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_6114_restrict__map__upds,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_real] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ D4 )
       => ( ( restri7804378014469731632T_real @ ( map_up4960461728960030954T_real @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_up4960461728960030954T_real @ ( restri7804378014469731632T_real @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6115_restrict__map__upds,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_o] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ D4 )
       => ( ( restri35548291844928020VEBT_o @ ( map_upds_VEBT_VEBT_o @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_VEBT_VEBT_o @ ( restri35548291844928020VEBT_o @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6116_restrict__map__upds,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ D4 )
       => ( ( restri774867724463461460BT_nat @ ( map_up889340208046081294BT_nat @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_up889340208046081294BT_nat @ ( restri774867724463461460BT_nat @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6117_restrict__map__upds,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int,D4: set_VEBT_VEBT,M: vEBT_VEBT > option_int] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ D4 )
       => ( ( restri772377253954411184BT_int @ ( map_up886849737537031018BT_int @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_up886849737537031018BT_int @ ( restri772377253954411184BT_int @ M @ ( minus_5127226145743854075T_VEBT @ D4 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6118_restrict__map__upds,axiom,
    ! [Xs2: list_real,Ys: list_real,D4: set_real,M: real > option_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ D4 )
       => ( ( restri4420043737358336266l_real @ ( map_upds_real_real @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_real_real @ ( restri4420043737358336266l_real @ M @ ( minus_minus_set_real @ D4 @ ( set_real2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6119_restrict__map__upds,axiom,
    ! [Xs2: list_real,Ys: list_o,D4: set_real,M: real > option_o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ D4 )
       => ( ( restrict_map_real_o @ ( map_upds_real_o @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_real_o @ ( restrict_map_real_o @ M @ ( minus_minus_set_real @ D4 @ ( set_real2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6120_restrict__map__upds,axiom,
    ! [Xs2: list_real,Ys: list_nat,D4: set_real,M: real > option_nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ D4 )
       => ( ( restri6827137924477938990al_nat @ ( map_upds_real_nat @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_real_nat @ ( restri6827137924477938990al_nat @ M @ ( minus_minus_set_real @ D4 @ ( set_real2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6121_restrict__map__upds,axiom,
    ! [Xs2: list_real,Ys: list_int,D4: set_real,M: real > option_int] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ D4 )
       => ( ( restri6824647453968888714al_int @ ( map_upds_real_int @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_real_int @ ( restri6824647453968888714al_int @ M @ ( minus_minus_set_real @ D4 @ ( set_real2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6122_restrict__map__upds,axiom,
    ! [Xs2: list_o,Ys: list_real,D4: set_o,M: $o > option_real] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ D4 )
       => ( ( restrict_map_o_real @ ( map_upds_o_real @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_o_real @ ( restrict_map_o_real @ M @ ( minus_minus_set_o @ D4 @ ( set_o2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6123_restrict__map__upds,axiom,
    ! [Xs2: list_o,Ys: list_o,D4: set_o,M: $o > option_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ D4 )
       => ( ( restrict_map_o_o @ ( map_upds_o_o @ M @ Xs2 @ Ys ) @ D4 )
          = ( map_upds_o_o @ ( restrict_map_o_o @ M @ ( minus_minus_set_o @ D4 @ ( set_o2 @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).

% restrict_map_upds
thf(fact_6124_nth__image,axiom,
    ! [L: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_eq_nat @ L @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( image_nat_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_VEBT_VEBTi2 @ ( take_VEBT_VEBTi @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6125_nth__image,axiom,
    ! [L: nat,Xs2: list_set_nat] :
      ( ( ord_less_eq_nat @ L @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( ( image_nat_set_nat @ ( nth_set_nat @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_set_nat2 @ ( take_set_nat @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6126_nth__image,axiom,
    ! [L: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ L @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( image_nat_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6127_nth__image,axiom,
    ! [L: nat,Xs2: list_real] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_real @ Xs2 ) )
     => ( ( image_nat_real @ ( nth_real @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_real2 @ ( take_real @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6128_nth__image,axiom,
    ! [L: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_o @ Xs2 ) )
     => ( ( image_nat_o @ ( nth_o @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_o2 @ ( take_o @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6129_nth__image,axiom,
    ! [L: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_nat @ Xs2 ) )
     => ( ( image_nat_nat @ ( nth_nat @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_nat2 @ ( take_nat @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6130_nth__image,axiom,
    ! [L: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ L @ ( size_size_list_int @ Xs2 ) )
     => ( ( image_nat_int @ ( nth_int @ Xs2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ L ) )
        = ( set_int2 @ ( take_int @ L @ Xs2 ) ) ) ) ).

% nth_image
thf(fact_6131_restrict__upd__same,axiom,
    ! [M: vEBT_VEBT > option_nat,X4: vEBT_VEBT,Y: nat] :
      ( ( restri774867724463461460BT_nat @ ( fun_up5885881570350532375on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( restri774867724463461460BT_nat @ M @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% restrict_upd_same
thf(fact_6132_restrict__upd__same,axiom,
    ! [M: vEBT_VEBT > option_num,X4: vEBT_VEBT,Y: num] :
      ( ( restri6555571547474015902BT_num @ ( fun_up6594125507299370081on_num @ M @ X4 @ ( some_num @ Y ) ) @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( restri6555571547474015902BT_num @ M @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% restrict_upd_same
thf(fact_6133_restrict__upd__same,axiom,
    ! [M: real > option_nat,X4: real,Y: nat] :
      ( ( restri6827137924477938990al_nat @ ( fun_up6677080212936659659on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
      = ( restri6827137924477938990al_nat @ M @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% restrict_upd_same
thf(fact_6134_restrict__upd__same,axiom,
    ! [M: real > option_num,X4: real,Y: num] :
      ( ( restri3384469710633717624al_num @ ( fun_up7385324149885497365on_num @ M @ X4 @ ( some_num @ Y ) ) @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
      = ( restri3384469710633717624al_num @ M @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% restrict_upd_same
thf(fact_6135_restrict__upd__same,axiom,
    ! [M: $o > option_nat,X4: $o,Y: nat] :
      ( ( restrict_map_o_nat @ ( fun_upd_o_option_nat @ M @ X4 @ ( some_nat @ Y ) ) @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
      = ( restrict_map_o_nat @ M @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% restrict_upd_same
thf(fact_6136_restrict__upd__same,axiom,
    ! [M: $o > option_num,X4: $o,Y: num] :
      ( ( restrict_map_o_num @ ( fun_upd_o_option_num @ M @ X4 @ ( some_num @ Y ) ) @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
      = ( restrict_map_o_num @ M @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% restrict_upd_same
thf(fact_6137_restrict__upd__same,axiom,
    ! [M: nat > option_nat,X4: nat,Y: nat] :
      ( ( restrict_map_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
      = ( restrict_map_nat_nat @ M @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% restrict_upd_same
thf(fact_6138_restrict__upd__same,axiom,
    ! [M: nat > option_num,X4: nat,Y: num] :
      ( ( restrict_map_nat_num @ ( fun_up2201401324907169337on_num @ M @ X4 @ ( some_num @ Y ) ) @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
      = ( restrict_map_nat_num @ M @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% restrict_upd_same
thf(fact_6139_restrict__upd__same,axiom,
    ! [M: int > option_nat,X4: int,Y: nat] :
      ( ( restrict_map_int_nat @ ( fun_up3620524117960394059on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
      = ( restrict_map_int_nat @ M @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% restrict_upd_same
thf(fact_6140_restrict__upd__same,axiom,
    ! [M: int > option_num,X4: int,Y: num] :
      ( ( restrict_map_int_num @ ( fun_up4328768054909231765on_num @ M @ X4 @ ( some_num @ Y ) ) @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
      = ( restrict_map_int_num @ M @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% restrict_upd_same
thf(fact_6141_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_6142_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_6143_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_6144_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_6145_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6146_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6147_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6148_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6149_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_6150_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_6151_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_6152_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_6153_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_6154_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_6155_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_6156_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_6157_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_6158_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_6159_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_6160_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_6161_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_6162_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_6163_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_6164_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_6165_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_6166_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_6167_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_6168_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_6169_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_6170_compl__le__compl__iff,axiom,
    ! [X4: assn,Y: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ X4 ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_eq_assn @ Y @ X4 ) ) ).

% compl_le_compl_iff
thf(fact_6171_compl__le__compl__iff,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X4 ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X4 ) ) ).

% compl_le_compl_iff
thf(fact_6172_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_6173_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_6174_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_6175_compl__less__compl__iff,axiom,
    ! [X4: assn,Y: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ X4 ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_assn @ Y @ X4 ) ) ).

% compl_less_compl_iff
thf(fact_6176_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_6177_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_6178_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_6179_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_6180_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_6181_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_6182_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_6183_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_6184_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_6185_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_6186_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_6187_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_6188_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_6189_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_6190_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_6191_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_6192_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_6193_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_6194_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_6195_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_6196_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_6197_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_6198_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_6199_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_6200_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_6201_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_6202_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_6203_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_6204_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6205_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6206_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6207_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6208_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_6209_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_6210_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_6211_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_6212_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_6213_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6214_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6215_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6216_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6217_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6218_Compl__anti__mono,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B4 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% Compl_anti_mono
thf(fact_6219_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B4 ) )
      = ( ord_less_eq_set_int @ B4 @ A3 ) ) ).

% Compl_subset_Compl_iff
thf(fact_6220_image__map__upd,axiom,
    ! [X4: nat,A3: set_nat,M: nat > option_nat,Y: nat] :
      ( ~ ( member_nat @ X4 @ A3 )
     => ( ( image_nat_option_nat @ ( fun_up1493157387958331631on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ A3 )
        = ( image_nat_option_nat @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6221_image__map__upd,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,M: vEBT_VEBT > option_nat,Y: nat] :
      ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( image_8844776943898047887on_nat @ ( fun_up5885881570350532375on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ A3 )
        = ( image_8844776943898047887on_nat @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6222_image__map__upd,axiom,
    ! [X4: real,A3: set_real,M: real > option_nat,Y: nat] :
      ( ~ ( member_real @ X4 @ A3 )
     => ( ( image_5574646975249130067on_nat @ ( fun_up6677080212936659659on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ A3 )
        = ( image_5574646975249130067on_nat @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6223_image__map__upd,axiom,
    ! [X4: int,A3: set_int,M: int > option_nat,Y: nat] :
      ( ~ ( member_int @ X4 @ A3 )
     => ( ( image_int_option_nat @ ( fun_up3620524117960394059on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ A3 )
        = ( image_int_option_nat @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6224_image__map__upd,axiom,
    ! [X4: nat,A3: set_nat,M: nat > option_num,Y: num] :
      ( ~ ( member_nat @ X4 @ A3 )
     => ( ( image_nat_option_num @ ( fun_up2201401324907169337on_num @ M @ X4 @ ( some_num @ Y ) ) @ A3 )
        = ( image_nat_option_num @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6225_image__map__upd,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT,M: vEBT_VEBT > option_num,Y: num] :
      ( ~ ( member_VEBT_VEBT @ X4 @ A3 )
     => ( ( image_329648843992109785on_num @ ( fun_up6594125507299370081on_num @ M @ X4 @ ( some_num @ Y ) ) @ A3 )
        = ( image_329648843992109785on_num @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6226_image__map__upd,axiom,
    ! [X4: real,A3: set_real,M: real > option_num,Y: num] :
      ( ~ ( member_real @ X4 @ A3 )
     => ( ( image_6282890912197967773on_num @ ( fun_up7385324149885497365on_num @ M @ X4 @ ( some_num @ Y ) ) @ A3 )
        = ( image_6282890912197967773on_num @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6227_image__map__upd,axiom,
    ! [X4: int,A3: set_int,M: int > option_num,Y: num] :
      ( ~ ( member_int @ X4 @ A3 )
     => ( ( image_int_option_num @ ( fun_up4328768054909231765on_num @ M @ X4 @ ( some_num @ Y ) ) @ A3 )
        = ( image_int_option_num @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6228_image__map__upd,axiom,
    ! [X4: set_nat,A3: set_set_nat,M: set_nat > option_nat,Y: nat] :
      ( ~ ( member_set_nat @ X4 @ A3 )
     => ( ( image_2753855177019846445on_nat @ ( fun_up5507366039599922085on_nat @ M @ X4 @ ( some_nat @ Y ) ) @ A3 )
        = ( image_2753855177019846445on_nat @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6229_image__map__upd,axiom,
    ! [X4: set_nat,A3: set_set_nat,M: set_nat > option_num,Y: num] :
      ( ~ ( member_set_nat @ X4 @ A3 )
     => ( ( image_3462099113968684151on_num @ ( fun_up6215609976548759791on_num @ M @ X4 @ ( some_num @ Y ) ) @ A3 )
        = ( image_3462099113968684151on_num @ M @ A3 ) ) ) ).

% image_map_upd
thf(fact_6230_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_6231_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_6232_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_6233_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_6234_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_6235_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_6236_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_6237_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_6238_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_6239_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_6240_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_6241_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_6242_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_6243_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_6244_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_6245_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_6246_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_6247_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_6248_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_6249_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_6250_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_6251_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_6252_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_6253_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_6254_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_6255_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_6256_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_6257_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_6258_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_6259_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_6260_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_6261_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_6262_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_6263_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_6264_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_6265_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_6266_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6267_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_6268_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_6269_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_6270_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_6271_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_6272_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_6273_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_6274_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_6275_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_6276_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_6277_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_6278_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_6279_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_6280_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_6281_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_6282_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_6283_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_6284_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_6285_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_6286_take__Suc__Cons,axiom,
    ! [N: nat,X4: $o,Xs2: list_o] :
      ( ( take_o @ ( suc @ N ) @ ( cons_o @ X4 @ Xs2 ) )
      = ( cons_o @ X4 @ ( take_o @ N @ Xs2 ) ) ) ).

% take_Suc_Cons
thf(fact_6287_take__Suc__Cons,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X4 @ Xs2 ) )
      = ( cons_nat @ X4 @ ( take_nat @ N @ Xs2 ) ) ) ).

% take_Suc_Cons
thf(fact_6288_take__Suc__Cons,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( take_int @ ( suc @ N ) @ ( cons_int @ X4 @ Xs2 ) )
      = ( cons_int @ X4 @ ( take_int @ N @ Xs2 ) ) ) ).

% take_Suc_Cons
thf(fact_6289_take__all,axiom,
    ! [Xs2: list_real,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N )
     => ( ( take_real @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_6290_take__all,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N )
     => ( ( take_o @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_6291_take__all,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
     => ( ( take_nat @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_6292_take__all,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N )
     => ( ( take_int @ N @ Xs2 )
        = Xs2 ) ) ).

% take_all
thf(fact_6293_take__all__iff,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ( take_real @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_6294_take__all__iff,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ( take_o @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_6295_take__all__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( take_nat @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_6296_take__all__iff,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ( take_int @ N @ Xs2 )
        = Xs2 )
      = ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).

% take_all_iff
thf(fact_6297_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBTi @ ( take_VEBT_VEBTi @ N @ Xs2 ) @ I )
        = ( nth_VEBT_VEBTi @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_6298_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ Xs2 ) @ I )
        = ( nth_VEBT_VEBT @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_6299_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( take_int @ N @ Xs2 ) @ I )
        = ( nth_int @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_6300_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs2 ) @ I )
        = ( nth_nat @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_6301_nth__take,axiom,
    ! [I: nat,N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_o @ ( take_o @ N @ Xs2 ) @ I )
        = ( nth_o @ Xs2 @ I ) ) ) ).

% nth_take
thf(fact_6302_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_nat,Y: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_nat @ N @ ( list_update_nat @ Xs2 @ M @ Y ) )
        = ( take_nat @ N @ Xs2 ) ) ) ).

% take_update_cancel
thf(fact_6303_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_o,Y: $o] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_o @ N @ ( list_update_o @ Xs2 @ M @ Y ) )
        = ( take_o @ N @ Xs2 ) ) ) ).

% take_update_cancel
thf(fact_6304_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_VEBT_VEBTi @ N @ ( list_u6098035379799741383_VEBTi @ Xs2 @ M @ Y ) )
        = ( take_VEBT_VEBTi @ N @ Xs2 ) ) ) ).

% take_update_cancel
thf(fact_6305_take__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( take_VEBT_VEBT @ N @ ( list_u1324408373059187874T_VEBT @ Xs2 @ M @ Y ) )
        = ( take_VEBT_VEBT @ N @ Xs2 ) ) ) ).

% take_update_cancel
thf(fact_6306_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = zero_zero_complex ) ).

% add_neg_numeral_special(8)
thf(fact_6307_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_6308_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_6309_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_6310_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% add_neg_numeral_special(7)
thf(fact_6311_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_6312_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_6313_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_6314_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_6315_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_6316_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_6317_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_6318_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6319_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6320_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6321_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6322_subset__Compl__singleton,axiom,
    ! [A3: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ~ ( member_VEBT_VEBT @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6323_subset__Compl__singleton,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6324_subset__Compl__singleton,axiom,
    ! [A3: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6325_subset__Compl__singleton,axiom,
    ! [A3: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6326_subset__Compl__singleton,axiom,
    ! [A3: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6327_subset__Compl__singleton,axiom,
    ! [A3: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_6328_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_real,I: nat,M: real > option_VEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( map_up4277394084060657369_VEBTi @ M @ Xs2 @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
        = ( map_up4277394084060657369_VEBTi @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6329_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_real,I: nat,M: real > option_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( map_up3774473461586936848T_VEBT @ M @ Xs2 @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
        = ( map_up3774473461586936848T_VEBT @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6330_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_o,I: nat,M: $o > option_VEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( map_up3026452068722249431_VEBTi @ M @ Xs2 @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
        = ( map_up3026452068722249431_VEBTi @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6331_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_o,I: nat,M: $o > option_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( map_upds_o_VEBT_VEBT @ M @ Xs2 @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
        = ( map_upds_o_VEBT_VEBT @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6332_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_nat,I: nat,M: nat > option_VEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( map_up547622734759332733_VEBTi @ M @ Xs2 @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
        = ( map_up547622734759332733_VEBTi @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6333_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_nat,I: nat,M: nat > option_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( map_up750602437722059884T_VEBT @ M @ Xs2 @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
        = ( map_up750602437722059884T_VEBT @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6334_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_int,I: nat,M: int > option_VEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( map_up2395700066158073305_VEBTi @ M @ Xs2 @ ( list_u6098035379799741383_VEBTi @ Ys @ I @ Y ) )
        = ( map_up2395700066158073305_VEBTi @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6335_map__upds__list__update2__drop,axiom,
    ! [Xs2: list_int,I: nat,M: int > option_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( map_up3480207007320876304T_VEBT @ M @ Xs2 @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
        = ( map_up3480207007320876304T_VEBT @ M @ Xs2 @ Ys ) ) ) ).

% map_upds_list_update2_drop
thf(fact_6336_map__upds__Cons,axiom,
    ! [M: nat > option_int,A: nat,As: list_nat,B: int,Bs: list_int] :
      ( ( map_upds_nat_int @ M @ ( cons_nat @ A @ As ) @ ( cons_int @ B @ Bs ) )
      = ( map_upds_nat_int @ ( fun_up6538678405303910731on_int @ M @ A @ ( some_int @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6337_map__upds__Cons,axiom,
    ! [M: int > option_int,A: int,As: list_int,B: int,Bs: list_int] :
      ( ( map_upds_int_int @ M @ ( cons_int @ A @ As ) @ ( cons_int @ B @ Bs ) )
      = ( map_upds_int_int @ ( fun_up8666045135305973159on_int @ M @ A @ ( some_int @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6338_map__upds__Cons,axiom,
    ! [M: nat > option4927543243414619207at_nat,A: nat,As: list_nat,B: product_prod_nat_nat,Bs: list_P6011104703257516679at_nat] :
      ( ( map_up1590851270032768825at_nat @ M @ ( cons_nat @ A @ As ) @ ( cons_P6512896166579812791at_nat @ B @ Bs ) )
      = ( map_up1590851270032768825at_nat @ ( fun_up2252230070437638584at_nat @ M @ A @ ( some_P7363390416028606310at_nat @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6339_map__upds__Cons,axiom,
    ! [M: int > option4927543243414619207at_nat,A: int,As: list_int,B: product_prod_nat_nat,Bs: list_P6011104703257516679at_nat] :
      ( ( map_up4628693354496902621at_nat @ M @ ( cons_int @ A @ As ) @ ( cons_P6512896166579812791at_nat @ B @ Bs ) )
      = ( map_up4628693354496902621at_nat @ ( fun_up1740315431458976348at_nat @ M @ A @ ( some_P7363390416028606310at_nat @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6340_map__upds__Cons,axiom,
    ! [M: nat > option_nat,A: nat,As: list_nat,B: nat,Bs: list_nat] :
      ( ( map_upds_nat_nat @ M @ ( cons_nat @ A @ As ) @ ( cons_nat @ B @ Bs ) )
      = ( map_upds_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6341_map__upds__Cons,axiom,
    ! [M: int > option_nat,A: int,As: list_int,B: nat,Bs: list_nat] :
      ( ( map_upds_int_nat @ M @ ( cons_int @ A @ As ) @ ( cons_nat @ B @ Bs ) )
      = ( map_upds_int_nat @ ( fun_up3620524117960394059on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6342_map__upds__Cons,axiom,
    ! [M: nat > option_num,A: nat,As: list_nat,B: num,Bs: list_num] :
      ( ( map_upds_nat_num @ M @ ( cons_nat @ A @ As ) @ ( cons_num @ B @ Bs ) )
      = ( map_upds_nat_num @ ( fun_up2201401324907169337on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6343_map__upds__Cons,axiom,
    ! [M: int > option_num,A: int,As: list_int,B: num,Bs: list_num] :
      ( ( map_upds_int_num @ M @ ( cons_int @ A @ As ) @ ( cons_num @ B @ Bs ) )
      = ( map_upds_int_num @ ( fun_up4328768054909231765on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs ) ) ).

% map_upds_Cons
thf(fact_6344_map__upds__twist,axiom,
    ! [A: int,As: list_int,M: int > option_nat,B: nat,Bs: list_nat] :
      ( ~ ( member_int @ A @ ( set_int2 @ As ) )
     => ( ( map_upds_int_nat @ ( fun_up3620524117960394059on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs )
        = ( fun_up3620524117960394059on_nat @ ( map_upds_int_nat @ M @ As @ Bs ) @ A @ ( some_nat @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6345_map__upds__twist,axiom,
    ! [A: vEBT_VEBT,As: list_VEBT_VEBT,M: vEBT_VEBT > option_nat,B: nat,Bs: list_nat] :
      ( ~ ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ As ) )
     => ( ( map_up889340208046081294BT_nat @ ( fun_up5885881570350532375on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs )
        = ( fun_up5885881570350532375on_nat @ ( map_up889340208046081294BT_nat @ M @ As @ Bs ) @ A @ ( some_nat @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6346_map__upds__twist,axiom,
    ! [A: nat,As: list_nat,M: nat > option_nat,B: nat,Bs: list_nat] :
      ( ~ ( member_nat @ A @ ( set_nat2 @ As ) )
     => ( ( map_upds_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs )
        = ( fun_up1493157387958331631on_nat @ ( map_upds_nat_nat @ M @ As @ Bs ) @ A @ ( some_nat @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6347_map__upds__twist,axiom,
    ! [A: real,As: list_real,M: real > option_nat,B: nat,Bs: list_nat] :
      ( ~ ( member_real @ A @ ( set_real2 @ As ) )
     => ( ( map_upds_real_nat @ ( fun_up6677080212936659659on_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs )
        = ( fun_up6677080212936659659on_nat @ ( map_upds_real_nat @ M @ As @ Bs ) @ A @ ( some_nat @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6348_map__upds__twist,axiom,
    ! [A: $o,As: list_o,M: $o > option_nat,B: nat,Bs: list_nat] :
      ( ~ ( member_o @ A @ ( set_o2 @ As ) )
     => ( ( map_upds_o_nat @ ( fun_upd_o_option_nat @ M @ A @ ( some_nat @ B ) ) @ As @ Bs )
        = ( fun_upd_o_option_nat @ ( map_upds_o_nat @ M @ As @ Bs ) @ A @ ( some_nat @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6349_map__upds__twist,axiom,
    ! [A: int,As: list_int,M: int > option_num,B: num,Bs: list_num] :
      ( ~ ( member_int @ A @ ( set_int2 @ As ) )
     => ( ( map_upds_int_num @ ( fun_up4328768054909231765on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs )
        = ( fun_up4328768054909231765on_num @ ( map_upds_int_num @ M @ As @ Bs ) @ A @ ( some_num @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6350_map__upds__twist,axiom,
    ! [A: vEBT_VEBT,As: list_VEBT_VEBT,M: vEBT_VEBT > option_num,B: num,Bs: list_num] :
      ( ~ ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ As ) )
     => ( ( map_up6670044031056635736BT_num @ ( fun_up6594125507299370081on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs )
        = ( fun_up6594125507299370081on_num @ ( map_up6670044031056635736BT_num @ M @ As @ Bs ) @ A @ ( some_num @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6351_map__upds__twist,axiom,
    ! [A: nat,As: list_nat,M: nat > option_num,B: num,Bs: list_num] :
      ( ~ ( member_nat @ A @ ( set_nat2 @ As ) )
     => ( ( map_upds_nat_num @ ( fun_up2201401324907169337on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs )
        = ( fun_up2201401324907169337on_num @ ( map_upds_nat_num @ M @ As @ Bs ) @ A @ ( some_num @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6352_map__upds__twist,axiom,
    ! [A: real,As: list_real,M: real > option_num,B: num,Bs: list_num] :
      ( ~ ( member_real @ A @ ( set_real2 @ As ) )
     => ( ( map_upds_real_num @ ( fun_up7385324149885497365on_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs )
        = ( fun_up7385324149885497365on_num @ ( map_upds_real_num @ M @ As @ Bs ) @ A @ ( some_num @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6353_map__upds__twist,axiom,
    ! [A: $o,As: list_o,M: $o > option_num,B: num,Bs: list_num] :
      ( ~ ( member_o @ A @ ( set_o2 @ As ) )
     => ( ( map_upds_o_num @ ( fun_upd_o_option_num @ M @ A @ ( some_num @ B ) ) @ As @ Bs )
        = ( fun_upd_o_option_num @ ( map_upds_o_num @ M @ As @ Bs ) @ A @ ( some_num @ B ) ) ) ) ).

% map_upds_twist
thf(fact_6354_sgn__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_6355_sgn__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_6356_sgn__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_6357_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_6358_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6359_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6360_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6361_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6362_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_6363_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_6364_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_6365_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_6366_Collect__neg__eq,axiom,
    ! [P: complex > $o] :
      ( ( collect_complex
        @ ^ [X: complex] :
            ~ ( P @ X ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6367_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ~ ( P @ X ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6368_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ~ ( P @ X ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6369_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ~ ( P @ X ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6370_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ~ ( P @ X ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6371_Compl__eq,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ~ ( member_VEBT_VEBT @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6372_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ~ ( member_real @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6373_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A5: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ~ ( member_complex @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6374_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ~ ( member_list_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6375_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ~ ( member_set_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6376_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ~ ( member_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6377_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ~ ( member_int @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_compl__mono,axiom,
    ! [X4: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X4 @ Y )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ ( uminus_uminus_assn @ X4 ) ) ) ).

% compl_mono
thf(fact_6388_compl__mono,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X4 ) ) ) ).

% compl_mono
thf(fact_6389_compl__le__swap1,axiom,
    ! [Y: assn,X4: assn] :
      ( ( ord_less_eq_assn @ Y @ ( uminus_uminus_assn @ X4 ) )
     => ( ord_less_eq_assn @ X4 @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_le_swap1
thf(fact_6390_compl__le__swap1,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X4 ) )
     => ( ord_less_eq_set_int @ X4 @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_6391_compl__le__swap2,axiom,
    ! [Y: assn,X4: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ X4 )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ X4 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_6392_compl__le__swap2,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X4 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X4 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_6393_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_6394_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_6395_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_6396_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_6397_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_6398_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_6399_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_6400_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_6401_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_6402_compl__less__swap1,axiom,
    ! [Y: assn,X4: assn] :
      ( ( ord_less_assn @ Y @ ( uminus_uminus_assn @ X4 ) )
     => ( ord_less_assn @ X4 @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_less_swap1
thf(fact_6403_compl__less__swap2,axiom,
    ! [Y: assn,X4: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ Y ) @ X4 )
     => ( ord_less_assn @ ( uminus_uminus_assn @ X4 ) @ Y ) ) ).

% compl_less_swap2
thf(fact_6404_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_6405_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_6406_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_6407_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_6408_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6409_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6410_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6411_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6412_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_6413_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_6414_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_6415_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_6416_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_6417_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_6418_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_6419_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_6420_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_6421_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_6422_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_6423_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_6424_abs__eq__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X4 )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6425_abs__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( abs_abs_real @ X4 )
        = ( abs_abs_real @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6426_abs__eq__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X4 )
        = ( abs_abs_rat @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6427_abs__eq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( abs_abs_int @ X4 )
        = ( abs_abs_int @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6428_Collect__imp__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_complex @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) @ ( collect_complex @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6429_Collect__imp__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6430_Collect__imp__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6431_Collect__imp__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6432_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6433_translation__subtract__Compl,axiom,
    ! [A: real,T: set_real] :
      ( ( image_real_real
        @ ^ [X: real] : ( minus_minus_real @ X @ A )
        @ ( uminus612125837232591019t_real @ T ) )
      = ( uminus612125837232591019t_real
        @ ( image_real_real
          @ ^ [X: real] : ( minus_minus_real @ X @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_6434_translation__subtract__Compl,axiom,
    ! [A: rat,T: set_rat] :
      ( ( image_rat_rat
        @ ^ [X: rat] : ( minus_minus_rat @ X @ A )
        @ ( uminus2201863774496077783et_rat @ T ) )
      = ( uminus2201863774496077783et_rat
        @ ( image_rat_rat
          @ ^ [X: rat] : ( minus_minus_rat @ X @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_6435_translation__subtract__Compl,axiom,
    ! [A: int,T: set_int] :
      ( ( image_int_int
        @ ^ [X: int] : ( minus_minus_int @ X @ A )
        @ ( uminus1532241313380277803et_int @ T ) )
      = ( uminus1532241313380277803et_int
        @ ( image_int_int
          @ ^ [X: int] : ( minus_minus_int @ X @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_6436_set__take__subset,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ N @ Xs2 ) ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_6437_set__take__subset,axiom,
    ! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_6438_set__take__subset,axiom,
    ! [N: nat,Xs2: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( take_real @ N @ Xs2 ) ) @ ( set_real2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_6439_set__take__subset,axiom,
    ! [N: nat,Xs2: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ N @ Xs2 ) ) @ ( set_o2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_6440_set__take__subset,axiom,
    ! [N: nat,Xs2: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ N @ Xs2 ) ) @ ( set_int2 @ Xs2 ) ) ).

% set_take_subset
thf(fact_6441_sorted__take,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( sorted_wrt_o @ ord_less_eq_o @ ( take_o @ N @ Xs2 ) ) ) ).

% sorted_take
thf(fact_6442_sorted__take,axiom,
    ! [Xs2: list_rat,N: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ ( take_rat @ N @ Xs2 ) ) ) ).

% sorted_take
thf(fact_6443_sorted__take,axiom,
    ! [Xs2: list_num,N: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( sorted_wrt_num @ ord_less_eq_num @ ( take_num @ N @ Xs2 ) ) ) ).

% sorted_take
thf(fact_6444_sorted__take,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs2 ) ) ) ).

% sorted_take
thf(fact_6445_sorted__take,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( take_int @ N @ Xs2 ) ) ) ).

% sorted_take
thf(fact_6446_None__notin__image__Some,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ~ ( member3954567711264315760at_nat @ none_P5556105721700978146at_nat @ ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6447_None__notin__image__Some,axiom,
    ! [A3: set_nat] :
      ~ ( member_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6448_None__notin__image__Some,axiom,
    ! [A3: set_num] :
      ~ ( member_option_num @ none_num @ ( image_num_option_num @ some_num @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6449_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_6450_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_6451_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_6452_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_6453_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6454_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_6455_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_6456_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_6457_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_6458_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_6459_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_6460_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_6461_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6462_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_6463_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_6464_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_6465_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6466_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_6467_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_6468_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_6469_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_6470_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_6471_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_6472_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_6473_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_6474_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_6475_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_6476_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_6477_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_6478_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_6479_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_6480_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_6481_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_6482_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_6483_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_6484_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_6485_square__eq__1__iff,axiom,
    ! [X4: complex] :
      ( ( ( times_times_complex @ X4 @ X4 )
        = one_one_complex )
      = ( ( X4 = one_one_complex )
        | ( X4
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_6486_square__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( times_times_real @ X4 @ X4 )
        = one_one_real )
      = ( ( X4 = one_one_real )
        | ( X4
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_6487_square__eq__1__iff,axiom,
    ! [X4: rat] :
      ( ( ( times_times_rat @ X4 @ X4 )
        = one_one_rat )
      = ( ( X4 = one_one_rat )
        | ( X4
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_6488_square__eq__1__iff,axiom,
    ! [X4: int] :
      ( ( ( times_times_int @ X4 @ X4 )
        = one_one_int )
      = ( ( X4 = one_one_int )
        | ( X4
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_6489_group__cancel_Osub2,axiom,
    ! [B4: complex,K: complex,B: complex,A: complex] :
      ( ( B4
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6490_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_6491_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_6492_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_6493_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6494_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6495_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6496_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6497_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6498_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6499_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6500_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6501_map__upd__upds__conv__if,axiom,
    ! [X4: int,Ys: list_num,Xs2: list_int,F: int > option_num,Y: num] :
      ( ( ( member_int @ X4 @ ( set_int2 @ ( take_int @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_int_num @ ( fun_up4328768054909231765on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_int_num @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_int @ X4 @ ( set_int2 @ ( take_int @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_int_num @ ( fun_up4328768054909231765on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up4328768054909231765on_num @ ( map_upds_int_num @ F @ Xs2 @ Ys ) @ X4 @ ( some_num @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6502_map__upd__upds__conv__if,axiom,
    ! [X4: vEBT_VEBT,Ys: list_num,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > option_num,Y: num] :
      ( ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_up6670044031056635736BT_num @ ( fun_up6594125507299370081on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( map_up6670044031056635736BT_num @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_up6670044031056635736BT_num @ ( fun_up6594125507299370081on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up6594125507299370081on_num @ ( map_up6670044031056635736BT_num @ F @ Xs2 @ Ys ) @ X4 @ ( some_num @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6503_map__upd__upds__conv__if,axiom,
    ! [X4: nat,Ys: list_num,Xs2: list_nat,F: nat > option_num,Y: num] :
      ( ( ( member_nat @ X4 @ ( set_nat2 @ ( take_nat @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_nat_num @ ( fun_up2201401324907169337on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_nat_num @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_nat @ X4 @ ( set_nat2 @ ( take_nat @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_nat_num @ ( fun_up2201401324907169337on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up2201401324907169337on_num @ ( map_upds_nat_num @ F @ Xs2 @ Ys ) @ X4 @ ( some_num @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6504_map__upd__upds__conv__if,axiom,
    ! [X4: real,Ys: list_num,Xs2: list_real,F: real > option_num,Y: num] :
      ( ( ( member_real @ X4 @ ( set_real2 @ ( take_real @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_real_num @ ( fun_up7385324149885497365on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_real_num @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_real @ X4 @ ( set_real2 @ ( take_real @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_real_num @ ( fun_up7385324149885497365on_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up7385324149885497365on_num @ ( map_upds_real_num @ F @ Xs2 @ Ys ) @ X4 @ ( some_num @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6505_map__upd__upds__conv__if,axiom,
    ! [X4: $o,Ys: list_num,Xs2: list_o,F: $o > option_num,Y: num] :
      ( ( ( member_o @ X4 @ ( set_o2 @ ( take_o @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_o_num @ ( fun_upd_o_option_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_o_num @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_o @ X4 @ ( set_o2 @ ( take_o @ ( size_size_list_num @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_o_num @ ( fun_upd_o_option_num @ F @ X4 @ ( some_num @ Y ) ) @ Xs2 @ Ys )
          = ( fun_upd_o_option_num @ ( map_upds_o_num @ F @ Xs2 @ Ys ) @ X4 @ ( some_num @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6506_map__upd__upds__conv__if,axiom,
    ! [X4: int,Ys: list_real,Xs2: list_int,F: int > option_real,Y: real] :
      ( ( ( member_int @ X4 @ ( set_int2 @ ( take_int @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_int_real @ ( fun_up3065300406613234215n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_int_real @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_int @ X4 @ ( set_int2 @ ( take_int @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_int_real @ ( fun_up3065300406613234215n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up3065300406613234215n_real @ ( map_upds_int_real @ F @ Xs2 @ Ys ) @ X4 @ ( some_real @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6507_map__upd__upds__conv__if,axiom,
    ! [X4: vEBT_VEBT,Ys: list_real,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > option_real,Y: real] :
      ( ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_up4960461728960030954T_real @ ( fun_up5420679954143522291n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( map_up4960461728960030954T_real @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_up4960461728960030954T_real @ ( fun_up5420679954143522291n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up5420679954143522291n_real @ ( map_up4960461728960030954T_real @ F @ Xs2 @ Ys ) @ X4 @ ( some_real @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6508_map__upd__upds__conv__if,axiom,
    ! [X4: nat,Ys: list_real,Xs2: list_nat,F: nat > option_real,Y: real] :
      ( ( ( member_nat @ X4 @ ( set_nat2 @ ( take_nat @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_nat_real @ ( fun_up6877590261764547531n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_nat_real @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_nat @ X4 @ ( set_nat2 @ ( take_nat @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_nat_real @ ( fun_up6877590261764547531n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up6877590261764547531n_real @ ( map_upds_nat_real @ F @ Xs2 @ Ys ) @ X4 @ ( some_real @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6509_map__upd__upds__conv__if,axiom,
    ! [X4: real,Ys: list_real,Xs2: list_real,F: real > option_real,Y: real] :
      ( ( ( member_real @ X4 @ ( set_real2 @ ( take_real @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_real_real @ ( fun_up7779911115344551847n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_real_real @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_real @ X4 @ ( set_real2 @ ( take_real @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_real_real @ ( fun_up7779911115344551847n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up7779911115344551847n_real @ ( map_upds_real_real @ F @ Xs2 @ Ys ) @ X4 @ ( some_real @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6510_map__upd__upds__conv__if,axiom,
    ! [X4: $o,Ys: list_real,Xs2: list_o,F: $o > option_real,Y: real] :
      ( ( ( member_o @ X4 @ ( set_o2 @ ( take_o @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_o_real @ ( fun_up2635748934393463895n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( map_upds_o_real @ F @ Xs2 @ Ys ) ) )
      & ( ~ ( member_o @ X4 @ ( set_o2 @ ( take_o @ ( size_size_list_real @ Ys ) @ Xs2 ) ) )
       => ( ( map_upds_o_real @ ( fun_up2635748934393463895n_real @ F @ X4 @ ( some_real @ Y ) ) @ Xs2 @ Ys )
          = ( fun_up2635748934393463895n_real @ ( map_upds_o_real @ F @ Xs2 @ Ys ) @ X4 @ ( some_real @ Y ) ) ) ) ) ).

% map_upd_upds_conv_if
thf(fact_6511_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6512_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_6513_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_6514_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_6515_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6516_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6517_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6518_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6519_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6520_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6521_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6522_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6523_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6524_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6525_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6526_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6527_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6528_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6529_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6530_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6531_subset__Compl__self__eq,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_6532_subset__Compl__self__eq,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_6533_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_6534_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_6535_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
       != ( sgn_sgn_Code_integer @ A ) )
     => ( ( ( sgn_sgn_Code_integer @ A )
         != zero_z3403309356797280102nteger )
       => ( ( ( sgn_sgn_Code_integer @ B )
           != zero_z3403309356797280102nteger )
         => ( ( sgn_sgn_Code_integer @ A )
            = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6536_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6537_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
       != ( sgn_sgn_rat @ A ) )
     => ( ( ( sgn_sgn_rat @ A )
         != zero_zero_rat )
       => ( ( ( sgn_sgn_rat @ B )
           != zero_zero_rat )
         => ( ( sgn_sgn_rat @ A )
            = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6538_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6539_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_6540_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_6541_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_6542_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_6543_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_6544_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ M @ Xs2 ) ) @ ( set_VEBT_VEBT2 @ ( take_VEBT_VEBT @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_6545_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_6546_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_real @ ( set_real2 @ ( take_real @ M @ Xs2 ) ) @ ( set_real2 @ ( take_real @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_6547_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ M @ Xs2 ) ) @ ( set_o2 @ ( take_o @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_6548_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ M @ Xs2 ) ) @ ( set_int2 @ ( take_int @ N @ Xs2 ) ) ) ) ).

% set_take_subset_set_take
thf(fact_6549_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_6550_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_6551_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_6552_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_6553_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_6554_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_6555_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_6556_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_6557_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_6558_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_6559_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_6560_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_6561_in__image__insert__iff,axiom,
    ! [B4: set_set_VEBT_VEBT,X4: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ! [C6: set_VEBT_VEBT] :
          ( ( member_set_VEBT_VEBT @ C6 @ B4 )
         => ~ ( member_VEBT_VEBT @ X4 @ C6 ) )
     => ( ( member_set_VEBT_VEBT @ A3 @ ( image_1661326939266726661T_VEBT @ ( insert_VEBT_VEBT @ X4 ) @ B4 ) )
        = ( ( member_VEBT_VEBT @ X4 @ A3 )
          & ( member_set_VEBT_VEBT @ ( minus_5127226145743854075T_VEBT @ A3 @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6562_in__image__insert__iff,axiom,
    ! [B4: set_set_set_nat,X4: set_nat,A3: set_set_nat] :
      ( ! [C6: set_set_nat] :
          ( ( member_set_set_nat @ C6 @ B4 )
         => ~ ( member_set_nat @ X4 @ C6 ) )
     => ( ( member_set_set_nat @ A3 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X4 ) @ B4 ) )
        = ( ( member_set_nat @ X4 @ A3 )
          & ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6563_in__image__insert__iff,axiom,
    ! [B4: set_set_real,X4: real,A3: set_real] :
      ( ! [C6: set_real] :
          ( ( member_set_real @ C6 @ B4 )
         => ~ ( member_real @ X4 @ C6 ) )
     => ( ( member_set_real @ A3 @ ( image_2436557299294012491t_real @ ( insert_real @ X4 ) @ B4 ) )
        = ( ( member_real @ X4 @ A3 )
          & ( member_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6564_in__image__insert__iff,axiom,
    ! [B4: set_set_o,X4: $o,A3: set_o] :
      ( ! [C6: set_o] :
          ( ( member_set_o @ C6 @ B4 )
         => ~ ( member_o @ X4 @ C6 ) )
     => ( ( member_set_o @ A3 @ ( image_set_o_set_o @ ( insert_o @ X4 ) @ B4 ) )
        = ( ( member_o @ X4 @ A3 )
          & ( member_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6565_in__image__insert__iff,axiom,
    ! [B4: set_set_int,X4: int,A3: set_int] :
      ( ! [C6: set_int] :
          ( ( member_set_int @ C6 @ B4 )
         => ~ ( member_int @ X4 @ C6 ) )
     => ( ( member_set_int @ A3 @ ( image_524474410958335435et_int @ ( insert_int @ X4 ) @ B4 ) )
        = ( ( member_int @ X4 @ A3 )
          & ( member_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6566_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,X4: nat,A3: set_nat] :
      ( ! [C6: set_nat] :
          ( ( member_set_nat @ C6 @ B4 )
         => ~ ( member_nat @ X4 @ C6 ) )
     => ( ( member_set_nat @ A3 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X4 ) @ B4 ) )
        = ( ( member_nat @ X4 @ A3 )
          & ( member_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6567_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6568_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_6569_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_6570_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_6571_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6572_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6573_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6574_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6575_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6576_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6577_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6578_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6579_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_6580_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_6581_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_6582_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_6583_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6584_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_6585_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_6586_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_6587_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_6588_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_6589_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_6590_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_6591_Compl__insert,axiom,
    ! [X4: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ A3 ) )
      = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A3 ) @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Compl_insert
thf(fact_6592_Compl__insert,axiom,
    ! [X4: real,A3: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X4 @ A3 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A3 ) @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_6593_Compl__insert,axiom,
    ! [X4: $o,A3: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X4 @ A3 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A3 ) @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_6594_Compl__insert,axiom,
    ! [X4: int,A3: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ A3 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_6595_Compl__insert,axiom,
    ! [X4: nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ A3 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_6596_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_eq_nat @ K @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s7982070591426661849_VEBTi @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
         => ( ( take_VEBT_VEBTi @ K @ Xs2 )
            = ( take_VEBT_VEBTi @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6597_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ K @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
         => ( ( take_VEBT_VEBT @ K @ Xs2 )
            = ( take_VEBT_VEBT @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6598_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_real,Ys: list_real] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_real @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_real @ Xs2 @ I2 )
                = ( nth_real @ Ys @ I2 ) ) )
         => ( ( take_real @ K @ Xs2 )
            = ( take_real @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6599_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_o,Ys: list_o] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_o @ Xs2 @ I2 )
                = ( nth_o @ Ys @ I2 ) ) )
         => ( ( take_o @ K @ Xs2 )
            = ( take_o @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6600_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_nat @ Xs2 @ I2 )
                = ( nth_nat @ Ys @ I2 ) ) )
         => ( ( take_nat @ K @ Xs2 )
            = ( take_nat @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6601_nth__take__lemma,axiom,
    ! [K: nat,Xs2: list_int,Ys: list_int] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_int @ Xs2 @ I2 )
                = ( nth_int @ Ys @ I2 ) ) )
         => ( ( take_int @ K @ Xs2 )
            = ( take_int @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_6602_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_6603_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_6604_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_6605_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_6606_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_6607_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_6608_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_6609_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_6610_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6611_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6612_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6613_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6614_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6615_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6616_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6617_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6618_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6619_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6620_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6621_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6622_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6623_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6624_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6625_zip__replicate2,axiom,
    ! [Xs2: list_nat,N: nat,Y: vEBT_VEBT] :
      ( ( zip_nat_VEBT_VEBT @ Xs2 @ ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( map_na3584885621601055599T_VEBT
        @ ^ [X: nat] : ( produc599794634098209291T_VEBT @ X @ Y )
        @ ( take_nat @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6626_zip__replicate2,axiom,
    ! [Xs2: list_o,N: nat,Y: vEBT_VEBT] :
      ( ( zip_o_VEBT_VEBT @ Xs2 @ ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( map_o_8925299737569714927T_VEBT
        @ ^ [X: $o] : ( produc2982872950893828659T_VEBT @ X @ Y )
        @ ( take_o @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6627_zip__replicate2,axiom,
    ! [Xs2: list_nat,N: nat,Y: $o] :
      ( ( zip_nat_o @ Xs2 @ ( replicate_o @ N @ Y ) )
      = ( map_na6716429308333697747_nat_o
        @ ^ [X: nat] : ( product_Pair_nat_o @ X @ Y )
        @ ( take_nat @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6628_zip__replicate2,axiom,
    ! [Xs2: list_o,N: nat,Y: $o] :
      ( ( zip_o_o @ Xs2 @ ( replicate_o @ N @ Y ) )
      = ( map_o_3702434973371374163od_o_o
        @ ^ [X: $o] : ( product_Pair_o_o @ X @ Y )
        @ ( take_o @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6629_zip__replicate2,axiom,
    ! [Xs2: list_nat,N: nat,Y: nat] :
      ( ( zip_nat_nat @ Xs2 @ ( replicate_nat @ N @ Y ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ Y )
        @ ( take_nat @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6630_zip__replicate2,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,Y: nat] :
      ( ( zip_VEBT_VEBT_nat @ Xs2 @ ( replicate_nat @ N @ Y ) )
      = ( map_VE6666121349297532579BT_nat
        @ ^ [X: vEBT_VEBT] : ( produc738532404422230701BT_nat @ X @ Y )
        @ ( take_VEBT_VEBT @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6631_zip__replicate2,axiom,
    ! [Xs2: list_int,N: nat,Y: int] :
      ( ( zip_int_int @ Xs2 @ ( replicate_int @ N @ Y ) )
      = ( map_in7157766398909135175nt_int
        @ ^ [X: int] : ( product_Pair_int_int @ X @ Y )
        @ ( take_int @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6632_zip__replicate2,axiom,
    ! [Xs2: list_Code_integer,N: nat,Y: code_integer] :
      ( ( zip_Co3543743374963494515nteger @ Xs2 @ ( replic7707675349574490269nteger @ N @ Y ) )
      = ( map_Co3589949550033412536nteger
        @ ^ [X: code_integer] : ( produc1086072967326762835nteger @ X @ Y )
        @ ( take_Code_integer @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6633_zip__replicate2,axiom,
    ! [Xs2: list_nat,N: nat,Y: num] :
      ( ( zip_nat_num @ Xs2 @ ( replicate_num @ N @ Y ) )
      = ( map_na8006665559001981237at_num
        @ ^ [X: nat] : ( product_Pair_nat_num @ X @ Y )
        @ ( take_nat @ N @ Xs2 ) ) ) ).

% zip_replicate2
thf(fact_6634_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_6635_restrict__complement__singleton__eq,axiom,
    ! [F: vEBT_VEBT > option_nat,X4: vEBT_VEBT] :
      ( ( restri774867724463461460BT_nat @ F @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( fun_up5885881570350532375on_nat @ F @ X4 @ none_nat ) ) ).

% restrict_complement_singleton_eq
thf(fact_6636_restrict__complement__singleton__eq,axiom,
    ! [F: vEBT_VEBT > option_num,X4: vEBT_VEBT] :
      ( ( restri6555571547474015902BT_num @ F @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( fun_up6594125507299370081on_num @ F @ X4 @ none_num ) ) ).

% restrict_complement_singleton_eq
thf(fact_6637_restrict__complement__singleton__eq,axiom,
    ! [F: real > option_nat,X4: real] :
      ( ( restri6827137924477938990al_nat @ F @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
      = ( fun_up6677080212936659659on_nat @ F @ X4 @ none_nat ) ) ).

% restrict_complement_singleton_eq
thf(fact_6638_restrict__complement__singleton__eq,axiom,
    ! [F: real > option_num,X4: real] :
      ( ( restri3384469710633717624al_num @ F @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
      = ( fun_up7385324149885497365on_num @ F @ X4 @ none_num ) ) ).

% restrict_complement_singleton_eq
thf(fact_6639_restrict__complement__singleton__eq,axiom,
    ! [F: $o > option_nat,X4: $o] :
      ( ( restrict_map_o_nat @ F @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
      = ( fun_upd_o_option_nat @ F @ X4 @ none_nat ) ) ).

% restrict_complement_singleton_eq
thf(fact_6640_restrict__complement__singleton__eq,axiom,
    ! [F: $o > option_num,X4: $o] :
      ( ( restrict_map_o_num @ F @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
      = ( fun_upd_o_option_num @ F @ X4 @ none_num ) ) ).

% restrict_complement_singleton_eq
thf(fact_6641_restrict__complement__singleton__eq,axiom,
    ! [F: nat > option_nat,X4: nat] :
      ( ( restrict_map_nat_nat @ F @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
      = ( fun_up1493157387958331631on_nat @ F @ X4 @ none_nat ) ) ).

% restrict_complement_singleton_eq
thf(fact_6642_restrict__complement__singleton__eq,axiom,
    ! [F: nat > option_num,X4: nat] :
      ( ( restrict_map_nat_num @ F @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
      = ( fun_up2201401324907169337on_num @ F @ X4 @ none_num ) ) ).

% restrict_complement_singleton_eq
thf(fact_6643_restrict__complement__singleton__eq,axiom,
    ! [F: int > option_nat,X4: int] :
      ( ( restrict_map_int_nat @ F @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
      = ( fun_up3620524117960394059on_nat @ F @ X4 @ none_nat ) ) ).

% restrict_complement_singleton_eq
thf(fact_6644_restrict__complement__singleton__eq,axiom,
    ! [F: int > option_num,X4: int] :
      ( ( restrict_map_int_num @ F @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
      = ( fun_up4328768054909231765on_num @ F @ X4 @ none_num ) ) ).

% restrict_complement_singleton_eq
thf(fact_6645_map__upd__eq__restrict,axiom,
    ! [M: vEBT_VEBT > option_nat,X4: vEBT_VEBT] :
      ( ( fun_up5885881570350532375on_nat @ M @ X4 @ none_nat )
      = ( restri774867724463461460BT_nat @ M @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6646_map__upd__eq__restrict,axiom,
    ! [M: vEBT_VEBT > option_num,X4: vEBT_VEBT] :
      ( ( fun_up6594125507299370081on_num @ M @ X4 @ none_num )
      = ( restri6555571547474015902BT_num @ M @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6647_map__upd__eq__restrict,axiom,
    ! [M: real > option_nat,X4: real] :
      ( ( fun_up6677080212936659659on_nat @ M @ X4 @ none_nat )
      = ( restri6827137924477938990al_nat @ M @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6648_map__upd__eq__restrict,axiom,
    ! [M: real > option_num,X4: real] :
      ( ( fun_up7385324149885497365on_num @ M @ X4 @ none_num )
      = ( restri3384469710633717624al_num @ M @ ( uminus612125837232591019t_real @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6649_map__upd__eq__restrict,axiom,
    ! [M: $o > option_nat,X4: $o] :
      ( ( fun_upd_o_option_nat @ M @ X4 @ none_nat )
      = ( restrict_map_o_nat @ M @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6650_map__upd__eq__restrict,axiom,
    ! [M: $o > option_num,X4: $o] :
      ( ( fun_upd_o_option_num @ M @ X4 @ none_num )
      = ( restrict_map_o_num @ M @ ( uminus_uminus_set_o @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6651_map__upd__eq__restrict,axiom,
    ! [M: nat > option_nat,X4: nat] :
      ( ( fun_up1493157387958331631on_nat @ M @ X4 @ none_nat )
      = ( restrict_map_nat_nat @ M @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6652_map__upd__eq__restrict,axiom,
    ! [M: nat > option_num,X4: nat] :
      ( ( fun_up2201401324907169337on_num @ M @ X4 @ none_num )
      = ( restrict_map_nat_num @ M @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6653_map__upd__eq__restrict,axiom,
    ! [M: int > option_nat,X4: int] :
      ( ( fun_up3620524117960394059on_nat @ M @ X4 @ none_nat )
      = ( restrict_map_int_nat @ M @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6654_map__upd__eq__restrict,axiom,
    ! [M: int > option_num,X4: int] :
      ( ( fun_up4328768054909231765on_num @ M @ X4 @ none_num )
      = ( restrict_map_int_num @ M @ ( uminus1532241313380277803et_int @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% map_upd_eq_restrict
thf(fact_6655_pochhammer__absorb__comp,axiom,
    ! [R3: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R3 ) @ K ) )
      = ( times_times_complex @ R3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R3 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6656_pochhammer__absorb__comp,axiom,
    ! [R3: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K ) )
      = ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6657_pochhammer__absorb__comp,axiom,
    ! [R3: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R3 ) @ K ) )
      = ( times_times_real @ R3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R3 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6658_pochhammer__absorb__comp,axiom,
    ! [R3: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K ) )
      = ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6659_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_6660_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_6661_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6662_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6663_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6664_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6665_VEBT__internal_Oheight_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X4 )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Uu2 @ 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_6666_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_6667_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_6668_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_6669_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_6670_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_6671_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_6672_VEBT__internal_Oheight_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ 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 @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_6673_fold__union__pair,axiom,
    ! [B4: set_nat,X4: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ bot_bo2099793752762293965at_nat )
              @ B4 ) )
          @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_6674_fold__union__pair,axiom,
    ! [B4: set_nat,X4: vEBT_VEBT,A3: set_Pr7556676689462069481BT_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( sup_su6061789376821058069BT_nat
          @ ( comple9061401370350521660BT_nat
            @ ( image_2578358110157900224BT_nat
              @ ^ [Y4: nat] : ( insert8978894354669351395BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y4 ) @ bot_bo1642239108664514429BT_nat )
              @ B4 ) )
          @ A3 )
        = ( finite1392145723888408323BT_nat
          @ ^ [Y4: nat] : ( insert8978894354669351395BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y4 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_6675_fold__union__pair,axiom,
    ! [B4: set_int,X4: int,A3: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ B4 )
     => ( ( sup_su6024340866399070445nt_int
          @ ( comple5382143125604098054nt_int
            @ ( image_8635204845542730022nt_int
              @ ^ [Y4: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ bot_bo1796632182523588997nt_int )
              @ B4 ) )
          @ A3 )
        = ( finite5202366122487795491nt_int
          @ ^ [Y4: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_6676_fold__union__pair,axiom,
    ! [B4: set_Code_integer,X4: code_integer,A3: set_Pr4811707699266497531nteger] :
      ( ( finite6017078050557962740nteger @ B4 )
     => ( ( sup_su3575575879904067535nteger
          @ ( comple2203973573673791208nteger
            @ ( image_8529064383210301591nteger
              @ ^ [Y4: code_integer] : ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ X4 @ Y4 ) @ bot_bo4276436098303576167nteger )
              @ B4 ) )
          @ A3 )
        = ( finite6473783728727339668nteger
          @ ^ [Y4: code_integer] : ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ X4 @ Y4 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_6677_fold__union__pair,axiom,
    ! [B4: set_num,X4: nat,A3: set_Pr6200539531224447659at_num] :
      ( ( finite_finite_num @ B4 )
     => ( ( sup_su2042722026077122175at_num
          @ ( comple1400524285282149784at_num
            @ ( image_4778453735051108682at_num
              @ ^ [Y4: num] : ( insert8920054152555992091at_num @ ( product_Pair_nat_num @ X4 @ Y4 ) @ bot_bo7038385379056416535at_num )
              @ B4 ) )
          @ A3 )
        = ( finite1345615011996174151at_num
          @ ^ [Y4: num] : ( insert8920054152555992091at_num @ ( product_Pair_nat_num @ X4 @ Y4 ) )
          @ A3
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_6678_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_6679_real__add__minus__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X4 = A ) ) ).

% real_add_minus_iff
thf(fact_6680_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_6681_cSup__singleton,axiom,
    ! [X4: $o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ X4 @ bot_bot_set_o ) )
      = X4 ) ).

% cSup_singleton
thf(fact_6682_cSup__singleton,axiom,
    ! [X4: int] :
      ( ( complete_Sup_Sup_int @ ( insert_int @ X4 @ bot_bot_set_int ) )
      = X4 ) ).

% cSup_singleton
thf(fact_6683_cSup__singleton,axiom,
    ! [X4: nat] :
      ( ( complete_Sup_Sup_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
      = X4 ) ).

% cSup_singleton
thf(fact_6684_cSup__singleton,axiom,
    ! [X4: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
      = X4 ) ).

% cSup_singleton
thf(fact_6685_cSup__singleton,axiom,
    ! [X4: real] :
      ( ( comple1385675409528146559p_real @ ( insert_real @ X4 @ bot_bot_set_real ) )
      = X4 ) ).

% cSup_singleton
thf(fact_6686_cSup__atLeastLessThan,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ Y @ X4 )
     => ( ( comple1385675409528146559p_real @ ( set_or66887138388493659n_real @ Y @ X4 ) )
        = X4 ) ) ).

% cSup_atLeastLessThan
thf(fact_6687_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_6688_Max__singleton,axiom,
    ! [X4: real] :
      ( ( lattic4275903605611617917x_real @ ( insert_real @ X4 @ bot_bot_set_real ) )
      = X4 ) ).

% Max_singleton
thf(fact_6689_Max__singleton,axiom,
    ! [X4: $o] :
      ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X4 @ bot_bot_set_o ) )
      = X4 ) ).

% Max_singleton
thf(fact_6690_Max__singleton,axiom,
    ! [X4: nat] :
      ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
      = X4 ) ).

% Max_singleton
thf(fact_6691_Max__singleton,axiom,
    ! [X4: int] :
      ( ( lattic8263393255366662781ax_int @ ( insert_int @ X4 @ bot_bot_set_int ) )
      = X4 ) ).

% Max_singleton
thf(fact_6692_last__replicate,axiom,
    ! [N: nat,X4: nat] :
      ( ( N != zero_zero_nat )
     => ( ( last_nat @ ( replicate_nat @ N @ X4 ) )
        = X4 ) ) ).

% last_replicate
thf(fact_6693_last__replicate,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( last_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X4 ) )
        = X4 ) ) ).

% last_replicate
thf(fact_6694_last__replicate,axiom,
    ! [N: nat,X4: $o] :
      ( ( N != zero_zero_nat )
     => ( ( last_o @ ( replicate_o @ N @ X4 ) )
        = X4 ) ) ).

% last_replicate
thf(fact_6695_cSUP__const,axiom,
    ! [A3: set_nat,C: int] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_int
          @ ( image_nat_int
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6696_cSUP__const,axiom,
    ! [A3: set_int,C: int] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_int
          @ ( image_int_int
            @ ^ [X: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6697_cSUP__const,axiom,
    ! [A3: set_real,C: nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ( complete_Sup_Sup_nat
          @ ( image_real_nat
            @ ^ [X: real] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6698_cSUP__const,axiom,
    ! [A3: set_o,C: nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_nat
          @ ( image_o_nat
            @ ^ [X: $o] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6699_cSUP__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_nat
          @ ( image_nat_nat
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6700_cSUP__const,axiom,
    ! [A3: set_int,C: nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_nat
          @ ( image_int_nat
            @ ^ [X: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6701_cSUP__const,axiom,
    ! [A3: set_real,C: real] :
      ( ( A3 != bot_bot_set_real )
     => ( ( comple1385675409528146559p_real
          @ ( image_real_real
            @ ^ [X: real] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6702_cSUP__const,axiom,
    ! [A3: set_o,C: real] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple1385675409528146559p_real
          @ ( image_o_real
            @ ^ [X: $o] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6703_cSUP__const,axiom,
    ! [A3: set_nat,C: real] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple1385675409528146559p_real
          @ ( image_nat_real
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6704_cSUP__const,axiom,
    ! [A3: set_int,C: real] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple1385675409528146559p_real
          @ ( image_int_real
            @ ^ [X: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_6705_finite__UN__I,axiom,
    ! [A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_int] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( finite_finite_int @ ( B4 @ A4 ) ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6706_finite__UN__I,axiom,
    ! [A3: set_real,B4: real > set_int] :
      ( ( finite_finite_real @ A3 )
     => ( ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( finite_finite_int @ ( B4 @ A4 ) ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6707_finite__UN__I,axiom,
    ! [A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_complex] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( finite3207457112153483333omplex @ ( B4 @ A4 ) ) )
       => ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_7711551513561399379omplex @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6708_finite__UN__I,axiom,
    ! [A3: set_real,B4: real > set_complex] :
      ( ( finite_finite_real @ A3 )
     => ( ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( finite3207457112153483333omplex @ ( B4 @ A4 ) ) )
       => ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_2129611632225307415omplex @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6709_finite__UN__I,axiom,
    ! [A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_Code_integer] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( finite6017078050557962740nteger @ ( B4 @ A4 ) ) )
       => ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_7429379093677266050nteger @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6710_finite__UN__I,axiom,
    ! [A3: set_real,B4: real > set_Code_integer] :
      ( ( finite_finite_real @ A3 )
     => ( ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( finite6017078050557962740nteger @ ( B4 @ A4 ) ) )
       => ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_1541721294081005382nteger @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6711_finite__UN__I,axiom,
    ! [A3: set_nat,B4: nat > set_int] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite_finite_int @ ( B4 @ A4 ) ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6712_finite__UN__I,axiom,
    ! [A3: set_nat,B4: nat > set_complex] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite3207457112153483333omplex @ ( B4 @ A4 ) ) )
       => ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6713_finite__UN__I,axiom,
    ! [A3: set_nat,B4: nat > set_Code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite6017078050557962740nteger @ ( B4 @ A4 ) ) )
       => ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_2385391725447763818nteger @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6714_finite__UN__I,axiom,
    ! [A3: set_int,B4: int > set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( finite_finite_int @ ( B4 @ A4 ) ) )
       => ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B4 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_6715_Max_Obounded__iff,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A3 ) @ X4 )
          = ( ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
               => ( ord_le3102999989581377725nteger @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6716_Max_Obounded__iff,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A3 ) @ X4 )
          = ( ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ord_less_eq_real @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6717_Max_Obounded__iff,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X4 )
          = ( ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ord_less_eq_o @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6718_Max_Obounded__iff,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A3 ) @ X4 )
          = ( ! [X: rat] :
                ( ( member_rat @ X @ A3 )
               => ( ord_less_eq_rat @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6719_Max_Obounded__iff,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A3 ) @ X4 )
          = ( ! [X: num] :
                ( ( member_num @ X @ A3 )
               => ( ord_less_eq_num @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6720_Max_Obounded__iff,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X4 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A3 )
               => ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6721_Max_Obounded__iff,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X4 )
          = ( ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ord_less_eq_int @ X @ X4 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6722_Max__less__iff,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ ( lattic4901227151466704046nteger @ A3 ) @ X4 )
          = ( ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
               => ( ord_le6747313008572928689nteger @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6723_Max__less__iff,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X4 )
          = ( ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ord_less_o @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6724_Max__less__iff,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ord_less_real @ ( lattic4275903605611617917x_real @ A3 ) @ X4 )
          = ( ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ord_less_real @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6725_Max__less__iff,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ord_less_rat @ ( lattic7630753665789217321ax_rat @ A3 ) @ X4 )
          = ( ! [X: rat] :
                ( ( member_rat @ X @ A3 )
               => ( ord_less_rat @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6726_Max__less__iff,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ord_less_num @ ( lattic4823215512031491691ax_num @ A3 ) @ X4 )
          = ( ! [X: num] :
                ( ( member_num @ X @ A3 )
               => ( ord_less_num @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6727_Max__less__iff,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X4 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A3 )
               => ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6728_Max__less__iff,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X4 )
          = ( ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ord_less_int @ X @ X4 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6729_Max__const,axiom,
    ! [A3: set_nat,C: real] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic4275903605611617917x_real
            @ ( image_nat_real
              @ ^ [Uu3: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6730_Max__const,axiom,
    ! [A3: set_complex,C: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_complex_nat
              @ ^ [Uu3: complex] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6731_Max__const,axiom,
    ! [A3: set_Code_integer,C: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_951025933927791156er_nat
              @ ^ [Uu3: code_integer] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6732_Max__const,axiom,
    ! [A3: set_real,C: nat] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_real_nat
              @ ^ [Uu3: real] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6733_Max__const,axiom,
    ! [A3: set_o,C: nat] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_o_nat
              @ ^ [Uu3: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6734_Max__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_nat_nat
              @ ^ [Uu3: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6735_Max__const,axiom,
    ! [A3: set_int,C: nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( lattic8265883725875713057ax_nat
            @ ( image_int_nat
              @ ^ [Uu3: int] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6736_Max__const,axiom,
    ! [A3: set_complex,C: int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_complex_int
              @ ^ [Uu3: complex] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6737_Max__const,axiom,
    ! [A3: set_Code_integer,C: int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_948535463418740880er_int
              @ ^ [Uu3: code_integer] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6738_Max__const,axiom,
    ! [A3: set_real,C: int] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( lattic8263393255366662781ax_int
            @ ( image_real_int
              @ ^ [Uu3: real] : C
              @ A3 ) )
          = C ) ) ) ).

% Max_const
thf(fact_6739_set__concat,axiom,
    ! [Xs2: list_list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( concat_VEBT_VEBT @ Xs2 ) )
      = ( comple2820511241208326657T_VEBT @ ( image_6463372868993444447T_VEBT @ set_VEBT_VEBT2 @ ( set_list_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_6740_set__concat,axiom,
    ! [Xs2: list_list_real] :
      ( ( set_real2 @ ( concat_real @ Xs2 ) )
      = ( comple3096694443085538997t_real @ ( image_6239767680843085477t_real @ set_real2 @ ( set_list_real2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_6741_set__concat,axiom,
    ! [Xs2: list_list_o] :
      ( ( set_o2 @ ( concat_o @ Xs2 ) )
      = ( comple90263536869209701_set_o @ ( image_list_o_set_o @ set_o2 @ ( set_list_o2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_6742_set__concat,axiom,
    ! [Xs2: list_list_nat] :
      ( ( set_nat2 @ ( concat_nat @ Xs2 ) )
      = ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_6743_Some__Sup,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( some_o @ ( complete_Sup_Sup_o @ A3 ) )
        = ( comple4490649148004324043tion_o @ ( image_o_option_o @ some_o @ A3 ) ) ) ) ).

% Some_Sup
thf(fact_6744_Some__Sup,axiom,
    ! [A3: set_set_nat] :
      ( ( A3 != bot_bot_set_set_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ A3 ) )
        = ( comple8455683388168444585et_nat @ ( image_838307146230912995et_nat @ some_set_nat @ A3 ) ) ) ) ).

% Some_Sup
thf(fact_6745_cSup__eq__Max,axiom,
    ! [X7: set_real] :
      ( ( finite_finite_real @ X7 )
     => ( ( X7 != bot_bot_set_real )
       => ( ( comple1385675409528146559p_real @ X7 )
          = ( lattic4275903605611617917x_real @ X7 ) ) ) ) ).

% cSup_eq_Max
thf(fact_6746_cSup__eq__Max,axiom,
    ! [X7: set_nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( X7 != bot_bot_set_nat )
       => ( ( complete_Sup_Sup_nat @ X7 )
          = ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).

% cSup_eq_Max
thf(fact_6747_cSup__eq__Max,axiom,
    ! [X7: set_int] :
      ( ( finite_finite_int @ X7 )
     => ( ( X7 != bot_bot_set_int )
       => ( ( complete_Sup_Sup_int @ X7 )
          = ( lattic8263393255366662781ax_int @ X7 ) ) ) ) ).

% cSup_eq_Max
thf(fact_6748_cSup__eq__maximum,axiom,
    ! [Z: set_int,X7: set_set_int] :
      ( ( member_set_int @ Z @ X7 )
     => ( ! [X3: set_int] :
            ( ( member_set_int @ X3 @ X7 )
           => ( ord_less_eq_set_int @ X3 @ Z ) )
       => ( ( comple3221217463730067765et_int @ X7 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_6749_cSup__eq__maximum,axiom,
    ! [Z: int,X7: set_int] :
      ( ( member_int @ Z @ X7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X7 )
           => ( ord_less_eq_int @ X3 @ Z ) )
       => ( ( complete_Sup_Sup_int @ X7 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_6750_cSup__eq__maximum,axiom,
    ! [Z: nat,X7: set_nat] :
      ( ( member_nat @ Z @ X7 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( complete_Sup_Sup_nat @ X7 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_6751_cSup__eq__maximum,axiom,
    ! [Z: set_nat,X7: set_set_nat] :
      ( ( member_set_nat @ Z @ X7 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ X7 )
           => ( ord_less_eq_set_nat @ X3 @ Z ) )
       => ( ( comple7399068483239264473et_nat @ X7 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_6752_cSup__eq__maximum,axiom,
    ! [Z: real,X7: set_real] :
      ( ( member_real @ Z @ X7 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X7 )
           => ( ord_less_eq_real @ X3 @ Z ) )
       => ( ( comple1385675409528146559p_real @ X7 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_6753_cSup__eq,axiom,
    ! [X7: set_int,A: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ X7 )
         => ( ord_less_eq_int @ X3 @ A ) )
     => ( ! [Y3: int] :
            ( ! [X6: int] :
                ( ( member_int @ X6 @ X7 )
               => ( ord_less_eq_int @ X6 @ Y3 ) )
           => ( ord_less_eq_int @ A @ Y3 ) )
       => ( ( complete_Sup_Sup_int @ X7 )
          = A ) ) ) ).

% cSup_eq
thf(fact_6754_cSup__eq,axiom,
    ! [X7: set_real,A: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ X7 )
         => ( ord_less_eq_real @ X3 @ A ) )
     => ( ! [Y3: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ X7 )
               => ( ord_less_eq_real @ X6 @ Y3 ) )
           => ( ord_less_eq_real @ A @ Y3 ) )
       => ( ( comple1385675409528146559p_real @ X7 )
          = A ) ) ) ).

% cSup_eq
thf(fact_6755_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_6756_uminus__set__def,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( uminus2746543603091002386VEBT_o
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6757_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X: real] : ( member_real @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6758_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A5: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X: complex] : ( member_complex @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6759_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6760_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6761_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6762_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X: int] : ( member_int @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6763_Some__SUP,axiom,
    ! [A3: set_real,F: real > set_nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_2508973409438724873et_nat
            @ ^ [X: real] : ( some_set_nat @ ( F @ X ) )
            @ A3 ) ) ) ) ).

% Some_SUP
thf(fact_6764_Some__SUP,axiom,
    ! [A3: set_o,F: $o > set_nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_966425688229763753et_nat
            @ ^ [X: $o] : ( some_set_nat @ ( F @ X ) )
            @ A3 ) ) ) ) ).

% Some_SUP
thf(fact_6765_Some__SUP,axiom,
    ! [A3: set_nat,F: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_1085372205585808685et_nat
            @ ^ [X: nat] : ( some_set_nat @ ( F @ X ) )
            @ A3 ) ) ) ) ).

% Some_SUP
thf(fact_6766_Some__SUP,axiom,
    ! [A3: set_int,F: int > set_nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_2293489338514188681et_nat
            @ ^ [X: int] : ( some_set_nat @ ( F @ X ) )
            @ A3 ) ) ) ) ).

% Some_SUP
thf(fact_6767_cSup__eq__non__empty,axiom,
    ! [X7: set_o,A: $o] :
      ( ( X7 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X7 )
           => ( ord_less_eq_o @ X3 @ A ) )
       => ( ! [Y3: $o] :
              ( ! [X6: $o] :
                  ( ( member_o @ X6 @ X7 )
                 => ( ord_less_eq_o @ X6 @ Y3 ) )
             => ( ord_less_eq_o @ A @ Y3 ) )
         => ( ( complete_Sup_Sup_o @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6768_cSup__eq__non__empty,axiom,
    ! [X7: set_set_int,A: set_int] :
      ( ( X7 != bot_bot_set_set_int )
     => ( ! [X3: set_int] :
            ( ( member_set_int @ X3 @ X7 )
           => ( ord_less_eq_set_int @ X3 @ A ) )
       => ( ! [Y3: set_int] :
              ( ! [X6: set_int] :
                  ( ( member_set_int @ X6 @ X7 )
                 => ( ord_less_eq_set_int @ X6 @ Y3 ) )
             => ( ord_less_eq_set_int @ A @ Y3 ) )
         => ( ( comple3221217463730067765et_int @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6769_cSup__eq__non__empty,axiom,
    ! [X7: set_int,A: int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X7 )
           => ( ord_less_eq_int @ X3 @ A ) )
       => ( ! [Y3: int] :
              ( ! [X6: int] :
                  ( ( member_int @ X6 @ X7 )
                 => ( ord_less_eq_int @ X6 @ Y3 ) )
             => ( ord_less_eq_int @ A @ Y3 ) )
         => ( ( complete_Sup_Sup_int @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6770_cSup__eq__non__empty,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ A ) )
       => ( ! [Y3: nat] :
              ( ! [X6: nat] :
                  ( ( member_nat @ X6 @ X7 )
                 => ( ord_less_eq_nat @ X6 @ Y3 ) )
             => ( ord_less_eq_nat @ A @ Y3 ) )
         => ( ( complete_Sup_Sup_nat @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6771_cSup__eq__non__empty,axiom,
    ! [X7: set_set_nat,A: set_nat] :
      ( ( X7 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ X7 )
           => ( ord_less_eq_set_nat @ X3 @ A ) )
       => ( ! [Y3: set_nat] :
              ( ! [X6: set_nat] :
                  ( ( member_set_nat @ X6 @ X7 )
                 => ( ord_less_eq_set_nat @ X6 @ Y3 ) )
             => ( ord_less_eq_set_nat @ A @ Y3 ) )
         => ( ( comple7399068483239264473et_nat @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6772_cSup__eq__non__empty,axiom,
    ! [X7: set_real,A: real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X7 )
           => ( ord_less_eq_real @ X3 @ A ) )
       => ( ! [Y3: real] :
              ( ! [X6: real] :
                  ( ( member_real @ X6 @ X7 )
                 => ( ord_less_eq_real @ X6 @ Y3 ) )
             => ( ord_less_eq_real @ A @ Y3 ) )
         => ( ( comple1385675409528146559p_real @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_6773_cSup__least,axiom,
    ! [X7: set_o,Z: $o] :
      ( ( X7 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X7 )
           => ( ord_less_eq_o @ X3 @ Z ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6774_cSup__least,axiom,
    ! [X7: set_set_int,Z: set_int] :
      ( ( X7 != bot_bot_set_set_int )
     => ( ! [X3: set_int] :
            ( ( member_set_int @ X3 @ X7 )
           => ( ord_less_eq_set_int @ X3 @ Z ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6775_cSup__least,axiom,
    ! [X7: set_int,Z: int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X7 )
           => ( ord_less_eq_int @ X3 @ Z ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6776_cSup__least,axiom,
    ! [X7: set_nat,Z: nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6777_cSup__least,axiom,
    ! [X7: set_set_nat,Z: set_nat] :
      ( ( X7 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ X7 )
           => ( ord_less_eq_set_nat @ X3 @ Z ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6778_cSup__least,axiom,
    ! [X7: set_real,Z: real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X7 )
           => ( ord_less_eq_real @ X3 @ Z ) )
       => ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X7 ) @ Z ) ) ) ).

% cSup_least
thf(fact_6779_le__cSup__finite,axiom,
    ! [X7: set_set_int,X4: set_int] :
      ( ( finite6197958912794628473et_int @ X7 )
     => ( ( member_set_int @ X4 @ X7 )
       => ( ord_less_eq_set_int @ X4 @ ( comple3221217463730067765et_int @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_6780_le__cSup__finite,axiom,
    ! [X7: set_int,X4: int] :
      ( ( finite_finite_int @ X7 )
     => ( ( member_int @ X4 @ X7 )
       => ( ord_less_eq_int @ X4 @ ( complete_Sup_Sup_int @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_6781_le__cSup__finite,axiom,
    ! [X7: set_nat,X4: nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( member_nat @ X4 @ X7 )
       => ( ord_less_eq_nat @ X4 @ ( complete_Sup_Sup_nat @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_6782_le__cSup__finite,axiom,
    ! [X7: set_set_nat,X4: set_nat] :
      ( ( finite1152437895449049373et_nat @ X7 )
     => ( ( member_set_nat @ X4 @ X7 )
       => ( ord_less_eq_set_nat @ X4 @ ( comple7399068483239264473et_nat @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_6783_le__cSup__finite,axiom,
    ! [X7: set_real,X4: real] :
      ( ( finite_finite_real @ X7 )
     => ( ( member_real @ X4 @ X7 )
       => ( ord_less_eq_real @ X4 @ ( comple1385675409528146559p_real @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_6784_less__cSupE,axiom,
    ! [Y: int,X7: set_int] :
      ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X7 ) )
     => ( ( X7 != bot_bot_set_int )
       => ~ ! [X3: int] :
              ( ( member_int @ X3 @ X7 )
             => ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).

% less_cSupE
thf(fact_6785_less__cSupE,axiom,
    ! [Y: nat,X7: set_nat] :
      ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X7 ) )
     => ( ( X7 != bot_bot_set_nat )
       => ~ ! [X3: nat] :
              ( ( member_nat @ X3 @ X7 )
             => ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).

% less_cSupE
thf(fact_6786_less__cSupE,axiom,
    ! [Y: real,X7: set_real] :
      ( ( ord_less_real @ Y @ ( comple1385675409528146559p_real @ X7 ) )
     => ( ( X7 != bot_bot_set_real )
       => ~ ! [X3: real] :
              ( ( member_real @ X3 @ X7 )
             => ~ ( ord_less_real @ Y @ X3 ) ) ) ) ).

% less_cSupE
thf(fact_6787_less__cSupD,axiom,
    ! [X7: set_int,Z: int] :
      ( ( X7 != bot_bot_set_int )
     => ( ( ord_less_int @ Z @ ( complete_Sup_Sup_int @ X7 ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ X7 )
            & ( ord_less_int @ Z @ X3 ) ) ) ) ).

% less_cSupD
thf(fact_6788_less__cSupD,axiom,
    ! [X7: set_nat,Z: nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X7 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
            & ( ord_less_nat @ Z @ X3 ) ) ) ) ).

% less_cSupD
thf(fact_6789_less__cSupD,axiom,
    ! [X7: set_real,Z: real] :
      ( ( X7 != bot_bot_set_real )
     => ( ( ord_less_real @ Z @ ( comple1385675409528146559p_real @ X7 ) )
       => ? [X3: real] :
            ( ( member_real @ X3 @ X7 )
            & ( ord_less_real @ Z @ X3 ) ) ) ) ).

% less_cSupD
thf(fact_6790_finite__imp__Sup__less,axiom,
    ! [X7: set_int,X4: int,A: int] :
      ( ( finite_finite_int @ X7 )
     => ( ( member_int @ X4 @ X7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ X7 )
             => ( ord_less_int @ X3 @ A ) )
         => ( ord_less_int @ ( complete_Sup_Sup_int @ X7 ) @ A ) ) ) ) ).

% finite_imp_Sup_less
thf(fact_6791_finite__imp__Sup__less,axiom,
    ! [X7: set_nat,X4: nat,A: nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( member_nat @ X4 @ X7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ X7 )
             => ( ord_less_nat @ X3 @ A ) )
         => ( ord_less_nat @ ( complete_Sup_Sup_nat @ X7 ) @ A ) ) ) ) ).

% finite_imp_Sup_less
thf(fact_6792_finite__imp__Sup__less,axiom,
    ! [X7: set_real,X4: real,A: real] :
      ( ( finite_finite_real @ X7 )
     => ( ( member_real @ X4 @ X7 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ X7 )
             => ( ord_less_real @ X3 @ A ) )
         => ( ord_less_real @ ( comple1385675409528146559p_real @ X7 ) @ A ) ) ) ) ).

% finite_imp_Sup_less
thf(fact_6793_Max_OcoboundedI,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A @ A3 )
       => ( ord_less_eq_real @ A @ ( lattic4275903605611617917x_real @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6794_Max_OcoboundedI,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ A @ A3 )
       => ( ord_le3102999989581377725nteger @ A @ ( lattic4901227151466704046nteger @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6795_Max_OcoboundedI,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ( ord_less_eq_rat @ A @ ( lattic7630753665789217321ax_rat @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6796_Max_OcoboundedI,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ( ord_less_eq_num @ A @ ( lattic4823215512031491691ax_num @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6797_Max_OcoboundedI,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6798_Max_OcoboundedI,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ( ord_less_eq_int @ A @ ( lattic8263393255366662781ax_int @ A3 ) ) ) ) ).

% Max.coboundedI
thf(fact_6799_Max__eq__if,axiom,
    ! [A3: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B4 )
                  & ( ord_le3102999989581377725nteger @ X3 @ Xa2 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ B4 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ A3 )
                    & ( ord_le3102999989581377725nteger @ X3 @ Xa2 ) ) )
           => ( ( lattic4901227151466704046nteger @ A3 )
              = ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6800_Max__eq__if,axiom,
    ! [A3: set_rat,B4: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( finite_finite_rat @ B4 )
       => ( ! [X3: rat] :
              ( ( member_rat @ X3 @ A3 )
             => ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ B4 )
                  & ( ord_less_eq_rat @ X3 @ Xa2 ) ) )
         => ( ! [X3: rat] :
                ( ( member_rat @ X3 @ B4 )
               => ? [Xa2: rat] :
                    ( ( member_rat @ Xa2 @ A3 )
                    & ( ord_less_eq_rat @ X3 @ Xa2 ) ) )
           => ( ( lattic7630753665789217321ax_rat @ A3 )
              = ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6801_Max__eq__if,axiom,
    ! [A3: set_num,B4: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( finite_finite_num @ B4 )
       => ( ! [X3: num] :
              ( ( member_num @ X3 @ A3 )
             => ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ B4 )
                  & ( ord_less_eq_num @ X3 @ Xa2 ) ) )
         => ( ! [X3: num] :
                ( ( member_num @ X3 @ B4 )
               => ? [Xa2: num] :
                    ( ( member_num @ Xa2 @ A3 )
                    & ( ord_less_eq_num @ X3 @ Xa2 ) ) )
           => ( ( lattic4823215512031491691ax_num @ A3 )
              = ( lattic4823215512031491691ax_num @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6802_Max__eq__if,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( ord_less_eq_nat @ X3 @ Xa2 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ B4 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ A3 )
                    & ( ord_less_eq_nat @ X3 @ Xa2 ) ) )
           => ( ( lattic8265883725875713057ax_nat @ A3 )
              = ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6803_Max__eq__if,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B4 )
                  & ( ord_less_eq_int @ X3 @ Xa2 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ B4 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ A3 )
                    & ( ord_less_eq_int @ X3 @ Xa2 ) ) )
           => ( ( lattic8263393255366662781ax_int @ A3 )
              = ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6804_Max__eqI,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ A3 )
           => ( ord_less_eq_real @ Y3 @ X4 ) )
       => ( ( member_real @ X4 @ A3 )
         => ( ( lattic4275903605611617917x_real @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6805_Max__eqI,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [Y3: code_integer] :
            ( ( member_Code_integer @ Y3 @ A3 )
           => ( ord_le3102999989581377725nteger @ Y3 @ X4 ) )
       => ( ( member_Code_integer @ X4 @ A3 )
         => ( ( lattic4901227151466704046nteger @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6806_Max__eqI,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ! [Y3: rat] :
            ( ( member_rat @ Y3 @ A3 )
           => ( ord_less_eq_rat @ Y3 @ X4 ) )
       => ( ( member_rat @ X4 @ A3 )
         => ( ( lattic7630753665789217321ax_rat @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6807_Max__eqI,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ! [Y3: num] :
            ( ( member_num @ Y3 @ A3 )
           => ( ord_less_eq_num @ Y3 @ X4 ) )
       => ( ( member_num @ X4 @ A3 )
         => ( ( lattic4823215512031491691ax_num @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6808_Max__eqI,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A3 )
           => ( ord_less_eq_nat @ Y3 @ X4 ) )
       => ( ( member_nat @ X4 @ A3 )
         => ( ( lattic8265883725875713057ax_nat @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6809_Max__eqI,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ A3 )
           => ( ord_less_eq_int @ Y3 @ X4 ) )
       => ( ( member_int @ X4 @ A3 )
         => ( ( lattic8263393255366662781ax_int @ A3 )
            = X4 ) ) ) ) ).

% Max_eqI
thf(fact_6810_Max__ge,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X4 @ A3 )
       => ( ord_less_eq_real @ X4 @ ( lattic4275903605611617917x_real @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6811_Max__ge,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ X4 @ A3 )
       => ( ord_le3102999989581377725nteger @ X4 @ ( lattic4901227151466704046nteger @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6812_Max__ge,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ X4 @ A3 )
       => ( ord_less_eq_rat @ X4 @ ( lattic7630753665789217321ax_rat @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6813_Max__ge,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ X4 @ A3 )
       => ( ord_less_eq_num @ X4 @ ( lattic4823215512031491691ax_num @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6814_Max__ge,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X4 @ A3 )
       => ( ord_less_eq_nat @ X4 @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6815_Max__ge,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X4 @ A3 )
       => ( ord_less_eq_int @ X4 @ ( lattic8263393255366662781ax_int @ A3 ) ) ) ) ).

% Max_ge
thf(fact_6816_Max__in,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_6817_Max__in,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( member_real @ ( lattic4275903605611617917x_real @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_6818_Max__in,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( member_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_6819_Max__in,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_6820_Max__in,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( member_int @ ( lattic8263393255366662781ax_int @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_6821_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_6822_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_6823_real__minus__mult__self__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).

% real_minus_mult_self_le
thf(fact_6824_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_6825_cSUP__least,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > int,M8: int] :
      ( ( A3 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_VEBT_VEBT_int @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6826_cSUP__least,axiom,
    ! [A3: set_real,F: real > int,M8: int] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_real_int @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6827_cSUP__least,axiom,
    ! [A3: set_o,F: $o > int,M8: int] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6828_cSUP__least,axiom,
    ! [A3: set_nat,F: nat > int,M8: int] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6829_cSUP__least,axiom,
    ! [A3: set_int,F: int > int,M8: int] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6830_cSUP__least,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > nat,M8: nat] :
      ( ( A3 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_VEBT_VEBT_nat @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6831_cSUP__least,axiom,
    ! [A3: set_real,F: real > nat,M8: nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_real_nat @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6832_cSUP__least,axiom,
    ! [A3: set_o,F: $o > nat,M8: nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6833_cSUP__least,axiom,
    ! [A3: set_nat,F: nat > nat,M8: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6834_cSUP__least,axiom,
    ! [A3: set_int,F: int > nat,M8: nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M8 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A3 ) ) @ M8 ) ) ) ).

% cSUP_least
thf(fact_6835_finite__Sup__less__iff,axiom,
    ! [X7: set_int,A: int] :
      ( ( finite_finite_int @ X7 )
     => ( ( X7 != bot_bot_set_int )
       => ( ( ord_less_int @ ( complete_Sup_Sup_int @ X7 ) @ A )
          = ( ! [X: int] :
                ( ( member_int @ X @ X7 )
               => ( ord_less_int @ X @ A ) ) ) ) ) ) ).

% finite_Sup_less_iff
thf(fact_6836_finite__Sup__less__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( X7 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X7 ) @ A )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ X7 )
               => ( ord_less_nat @ X @ A ) ) ) ) ) ) ).

% finite_Sup_less_iff
thf(fact_6837_finite__Sup__less__iff,axiom,
    ! [X7: set_real,A: real] :
      ( ( finite_finite_real @ X7 )
     => ( ( X7 != bot_bot_set_real )
       => ( ( ord_less_real @ ( comple1385675409528146559p_real @ X7 ) @ A )
          = ( ! [X: real] :
                ( ( member_real @ X @ X7 )
               => ( ord_less_real @ X @ A ) ) ) ) ) ) ).

% finite_Sup_less_iff
thf(fact_6838_cSup__abs__le,axiom,
    ! [S3: set_int,A: int] :
      ( ( S3 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ S3 )
           => ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ A ) )
       => ( ord_less_eq_int @ ( abs_abs_int @ ( complete_Sup_Sup_int @ S3 ) ) @ A ) ) ) ).

% cSup_abs_le
thf(fact_6839_cSup__abs__le,axiom,
    ! [S3: set_real,A: real] :
      ( ( S3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ A ) )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( comple1385675409528146559p_real @ S3 ) ) @ A ) ) ) ).

% cSup_abs_le
thf(fact_6840_finite__Sup__in,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o,Y3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ( ( member_o @ Y3 @ A3 )
               => ( member_o @ ( sup_sup_o @ X3 @ Y3 ) @ A3 ) ) )
         => ( member_o @ ( complete_Sup_Sup_o @ A3 ) @ A3 ) ) ) ) ).

% finite_Sup_in
thf(fact_6841_finite__Sup__in,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y3: set_nat] :
              ( ( member_set_nat @ X3 @ A3 )
             => ( ( member_set_nat @ Y3 @ A3 )
               => ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ A3 ) ) )
         => ( member_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ A3 ) ) ) ) ).

% finite_Sup_in
thf(fact_6842_Max_OboundedI,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [A4: code_integer] :
              ( ( member_Code_integer @ A4 @ A3 )
             => ( ord_le3102999989581377725nteger @ A4 @ X4 ) )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6843_Max_OboundedI,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ A3 )
             => ( ord_less_eq_real @ A4 @ X4 ) )
         => ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6844_Max_OboundedI,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( ord_less_eq_o @ A4 @ X4 ) )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6845_Max_OboundedI,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ! [A4: rat] :
              ( ( member_rat @ A4 @ A3 )
             => ( ord_less_eq_rat @ A4 @ X4 ) )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6846_Max_OboundedI,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ! [A4: num] :
              ( ( member_num @ A4 @ A3 )
             => ( ord_less_eq_num @ A4 @ X4 ) )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6847_Max_OboundedI,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
             => ( ord_less_eq_nat @ A4 @ X4 ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6848_Max_OboundedI,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ A3 )
             => ( ord_less_eq_int @ A4 @ X4 ) )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X4 ) ) ) ) ).

% Max.boundedI
thf(fact_6849_Max_OboundedE,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A3 ) @ X4 )
         => ! [A11: code_integer] :
              ( ( member_Code_integer @ A11 @ A3 )
             => ( ord_le3102999989581377725nteger @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6850_Max_OboundedE,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A3 ) @ X4 )
         => ! [A11: real] :
              ( ( member_real @ A11 @ A3 )
             => ( ord_less_eq_real @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6851_Max_OboundedE,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X4 )
         => ! [A11: $o] :
              ( ( member_o @ A11 @ A3 )
             => ( ord_less_eq_o @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6852_Max_OboundedE,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A3 ) @ X4 )
         => ! [A11: rat] :
              ( ( member_rat @ A11 @ A3 )
             => ( ord_less_eq_rat @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6853_Max_OboundedE,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A3 ) @ X4 )
         => ! [A11: num] :
              ( ( member_num @ A11 @ A3 )
             => ( ord_less_eq_num @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6854_Max_OboundedE,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X4 )
         => ! [A11: nat] :
              ( ( member_nat @ A11 @ A3 )
             => ( ord_less_eq_nat @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6855_Max_OboundedE,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X4 )
         => ! [A11: int] :
              ( ( member_int @ A11 @ A3 )
             => ( ord_less_eq_int @ A11 @ X4 ) ) ) ) ) ).

% Max.boundedE
thf(fact_6856_eq__Max__iff,axiom,
    ! [A3: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( M
            = ( lattic4901227151466704046nteger @ A3 ) )
          = ( ( member_Code_integer @ M @ A3 )
            & ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
               => ( ord_le3102999989581377725nteger @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6857_eq__Max__iff,axiom,
    ! [A3: set_real,M: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( M
            = ( lattic4275903605611617917x_real @ A3 ) )
          = ( ( member_real @ M @ A3 )
            & ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ord_less_eq_real @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6858_eq__Max__iff,axiom,
    ! [A3: set_o,M: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( M
            = ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ( member_o @ M @ A3 )
            & ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ord_less_eq_o @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6859_eq__Max__iff,axiom,
    ! [A3: set_rat,M: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( M
            = ( lattic7630753665789217321ax_rat @ A3 ) )
          = ( ( member_rat @ M @ A3 )
            & ! [X: rat] :
                ( ( member_rat @ X @ A3 )
               => ( ord_less_eq_rat @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6860_eq__Max__iff,axiom,
    ! [A3: set_num,M: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( M
            = ( lattic4823215512031491691ax_num @ A3 ) )
          = ( ( member_num @ M @ A3 )
            & ! [X: num] :
                ( ( member_num @ X @ A3 )
               => ( ord_less_eq_num @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6861_eq__Max__iff,axiom,
    ! [A3: set_nat,M: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( M
            = ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ( member_nat @ M @ A3 )
            & ! [X: nat] :
                ( ( member_nat @ X @ A3 )
               => ( ord_less_eq_nat @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6862_eq__Max__iff,axiom,
    ! [A3: set_int,M: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( M
            = ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ( member_int @ M @ A3 )
            & ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ord_less_eq_int @ X @ M ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6863_Max__ge__iff,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ X4 @ ( lattic4901227151466704046nteger @ A3 ) )
          = ( ? [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
                & ( ord_le3102999989581377725nteger @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6864_Max__ge__iff,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ord_less_eq_real @ X4 @ ( lattic4275903605611617917x_real @ A3 ) )
          = ( ? [X: real] :
                ( ( member_real @ X @ A3 )
                & ( ord_less_eq_real @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6865_Max__ge__iff,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X4 @ ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ? [X: $o] :
                ( ( member_o @ X @ A3 )
                & ( ord_less_eq_o @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6866_Max__ge__iff,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ord_less_eq_rat @ X4 @ ( lattic7630753665789217321ax_rat @ A3 ) )
          = ( ? [X: rat] :
                ( ( member_rat @ X @ A3 )
                & ( ord_less_eq_rat @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6867_Max__ge__iff,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ord_less_eq_num @ X4 @ ( lattic4823215512031491691ax_num @ A3 ) )
          = ( ? [X: num] :
                ( ( member_num @ X @ A3 )
                & ( ord_less_eq_num @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6868_Max__ge__iff,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X4 @ ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6869_Max__ge__iff,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X4 @ ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ A3 )
                & ( ord_less_eq_int @ X4 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6870_Max__eq__iff,axiom,
    ! [A3: set_Code_integer,M: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ( lattic4901227151466704046nteger @ A3 )
            = M )
          = ( ( member_Code_integer @ M @ A3 )
            & ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
               => ( ord_le3102999989581377725nteger @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6871_Max__eq__iff,axiom,
    ! [A3: set_real,M: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ( lattic4275903605611617917x_real @ A3 )
            = M )
          = ( ( member_real @ M @ A3 )
            & ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ord_less_eq_real @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6872_Max__eq__iff,axiom,
    ! [A3: set_o,M: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ( lattic1921953407002678535_Max_o @ A3 )
            = M )
          = ( ( member_o @ M @ A3 )
            & ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ord_less_eq_o @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6873_Max__eq__iff,axiom,
    ! [A3: set_rat,M: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ( lattic7630753665789217321ax_rat @ A3 )
            = M )
          = ( ( member_rat @ M @ A3 )
            & ! [X: rat] :
                ( ( member_rat @ X @ A3 )
               => ( ord_less_eq_rat @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6874_Max__eq__iff,axiom,
    ! [A3: set_num,M: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ( lattic4823215512031491691ax_num @ A3 )
            = M )
          = ( ( member_num @ M @ A3 )
            & ! [X: num] :
                ( ( member_num @ X @ A3 )
               => ( ord_less_eq_num @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6875_Max__eq__iff,axiom,
    ! [A3: set_nat,M: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A3 )
            = M )
          = ( ( member_nat @ M @ A3 )
            & ! [X: nat] :
                ( ( member_nat @ X @ A3 )
               => ( ord_less_eq_nat @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6876_Max__eq__iff,axiom,
    ! [A3: set_int,M: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ( lattic8263393255366662781ax_int @ A3 )
            = M )
          = ( ( member_int @ M @ A3 )
            & ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ord_less_eq_int @ X @ M ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6877_Max__gr__iff,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ X4 @ ( lattic4901227151466704046nteger @ A3 ) )
          = ( ? [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
                & ( ord_le6747313008572928689nteger @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6878_Max__gr__iff,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ X4 @ ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ? [X: $o] :
                ( ( member_o @ X @ A3 )
                & ( ord_less_o @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6879_Max__gr__iff,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( ord_less_real @ X4 @ ( lattic4275903605611617917x_real @ A3 ) )
          = ( ? [X: real] :
                ( ( member_real @ X @ A3 )
                & ( ord_less_real @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6880_Max__gr__iff,axiom,
    ! [A3: set_rat,X4: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( ord_less_rat @ X4 @ ( lattic7630753665789217321ax_rat @ A3 ) )
          = ( ? [X: rat] :
                ( ( member_rat @ X @ A3 )
                & ( ord_less_rat @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6881_Max__gr__iff,axiom,
    ! [A3: set_num,X4: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( ord_less_num @ X4 @ ( lattic4823215512031491691ax_num @ A3 ) )
          = ( ? [X: num] :
                ( ( member_num @ X @ A3 )
                & ( ord_less_num @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6882_Max__gr__iff,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X4 @ ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6883_Max__gr__iff,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ X4 @ ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ A3 )
                & ( ord_less_int @ X4 @ X ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6884_Max__insert2,axiom,
    ! [A3: set_o,A: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ! [B3: $o] :
            ( ( member_o @ B3 @ A3 )
           => ( ord_less_eq_o @ B3 @ A ) )
       => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6885_Max__insert2,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ! [B3: real] :
            ( ( member_real @ B3 @ A3 )
           => ( ord_less_eq_real @ B3 @ A ) )
       => ( ( lattic4275903605611617917x_real @ ( insert_real @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6886_Max__insert2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [B3: code_integer] :
            ( ( member_Code_integer @ B3 @ A3 )
           => ( ord_le3102999989581377725nteger @ B3 @ A ) )
       => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6887_Max__insert2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ! [B3: rat] :
            ( ( member_rat @ B3 @ A3 )
           => ( ord_less_eq_rat @ B3 @ A ) )
       => ( ( lattic7630753665789217321ax_rat @ ( insert_rat @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6888_Max__insert2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ! [B3: num] :
            ( ( member_num @ B3 @ A3 )
           => ( ord_less_eq_num @ B3 @ A ) )
       => ( ( lattic4823215512031491691ax_num @ ( insert_num @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6889_Max__insert2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [B3: nat] :
            ( ( member_nat @ B3 @ A3 )
           => ( ord_less_eq_nat @ B3 @ A ) )
       => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6890_Max__insert2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ! [B3: int] :
            ( ( member_int @ B3 @ A3 )
           => ( ord_less_eq_int @ B3 @ A ) )
       => ( ( lattic8263393255366662781ax_int @ ( insert_int @ A @ A3 ) )
          = A ) ) ) ).

% Max_insert2
thf(fact_6891_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_6892_real__add__less__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_less_0_iff
thf(fact_6893_real__0__less__add__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_6894_real__0__le__add__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_6895_real__add__le__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_le_0_iff
thf(fact_6896_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_6897_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_real_def
thf(fact_6898_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_6899_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_6900_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_6901_finite__subset__Union,axiom,
    ! [A3: set_complex,B8: set_set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ ( comple8424636186594484919omplex @ B8 ) )
       => ~ ! [F8: set_set_complex] :
              ( ( finite6551019134538273531omplex @ F8 )
             => ( ( ord_le4750530260501030778omplex @ F8 @ B8 )
               => ~ ( ord_le211207098394363844omplex @ A3 @ ( comple8424636186594484919omplex @ F8 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_6902_finite__subset__Union,axiom,
    ! [A3: set_Code_integer,B8: set_set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ ( comple739944243200306918nteger @ B8 ) )
       => ~ ! [F8: set_set_Code_integer] :
              ( ( finite6931041176100689706nteger @ F8 )
             => ( ( ord_le1914454125413604393nteger @ F8 @ B8 )
               => ~ ( ord_le7084787975880047091nteger @ A3 @ ( comple739944243200306918nteger @ F8 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_6903_finite__subset__Union,axiom,
    ! [A3: set_int,B8: set_set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ A3 @ ( comple3221217463730067765et_int @ B8 ) )
       => ~ ! [F8: set_set_int] :
              ( ( finite6197958912794628473et_int @ F8 )
             => ( ( ord_le4403425263959731960et_int @ F8 @ B8 )
               => ~ ( ord_less_eq_set_int @ A3 @ ( comple3221217463730067765et_int @ F8 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_6904_finite__subset__Union,axiom,
    ! [A3: set_nat,B8: set_set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ B8 ) )
       => ~ ! [F8: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F8 )
             => ( ( ord_le6893508408891458716et_nat @ F8 @ B8 )
               => ~ ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ F8 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_6905_cSup__asclose,axiom,
    ! [S3: set_int,L: int,E2: int] :
      ( ( S3 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ S3 )
           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ L ) ) @ E2 ) )
       => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( complete_Sup_Sup_int @ S3 ) @ L ) ) @ E2 ) ) ) ).

% cSup_asclose
thf(fact_6906_cSup__asclose,axiom,
    ! [S3: set_real,L: real,E2: real] :
      ( ( S3 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ L ) ) @ E2 ) )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( comple1385675409528146559p_real @ S3 ) @ L ) ) @ E2 ) ) ) ).

% cSup_asclose
thf(fact_6907_Sup__fold__sup,axiom,
    ! [A3: set_set_real] :
      ( ( finite9007344921179782393t_real @ A3 )
     => ( ( comple3096694443085538997t_real @ A3 )
        = ( finite8257685287532064654t_real @ sup_sup_set_real @ bot_bot_set_real @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_6908_Sup__fold__sup,axiom,
    ! [A3: set_set_o] :
      ( ( finite_finite_set_o @ A3 )
     => ( ( comple90263536869209701_set_o @ A3 )
        = ( finite4337638375924247368_set_o @ sup_sup_set_o @ bot_bot_set_o @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_6909_Sup__fold__sup,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( comple3221217463730067765et_int @ A3 )
        = ( finite2508883933289561102et_int @ sup_sup_set_int @ bot_bot_set_int @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_6910_Sup__fold__sup,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( comple7399068483239264473et_nat @ A3 )
        = ( finite677925301803182934et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_6911_Max_Osubset__imp,axiom,
    ! [A3: set_Code_integer,B4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A3 @ B4 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ B4 )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A3 ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6912_Max_Osubset__imp,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( finite_finite_real @ B4 )
         => ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ A3 ) @ ( lattic4275903605611617917x_real @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6913_Max_Osubset__imp,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B4 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6914_Max_Osubset__imp,axiom,
    ! [A3: set_rat,B4: set_rat] :
      ( ( ord_less_eq_set_rat @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ B4 )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A3 ) @ ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6915_Max_Osubset__imp,axiom,
    ! [A3: set_num,B4: set_num] :
      ( ( ord_less_eq_set_num @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_num )
       => ( ( finite_finite_num @ B4 )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ A3 ) @ ( lattic4823215512031491691ax_num @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6916_Max_Osubset__imp,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6917_Max_Osubset__imp,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B4 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6918_Max__mono,axiom,
    ! [M8: set_Code_integer,N7: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ M8 @ N7 )
     => ( ( M8 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ N7 )
         => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ M8 ) @ ( lattic4901227151466704046nteger @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6919_Max__mono,axiom,
    ! [M8: set_real,N7: set_real] :
      ( ( ord_less_eq_set_real @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_real )
       => ( ( finite_finite_real @ N7 )
         => ( ord_less_eq_real @ ( lattic4275903605611617917x_real @ M8 ) @ ( lattic4275903605611617917x_real @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6920_Max__mono,axiom,
    ! [M8: set_o,N7: set_o] :
      ( ( ord_less_eq_set_o @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_o )
       => ( ( finite_finite_o @ N7 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M8 ) @ ( lattic1921953407002678535_Max_o @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6921_Max__mono,axiom,
    ! [M8: set_rat,N7: set_rat] :
      ( ( ord_less_eq_set_rat @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_rat )
       => ( ( finite_finite_rat @ N7 )
         => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ M8 ) @ ( lattic7630753665789217321ax_rat @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6922_Max__mono,axiom,
    ! [M8: set_num,N7: set_num] :
      ( ( ord_less_eq_set_num @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_num )
       => ( ( finite_finite_num @ N7 )
         => ( ord_less_eq_num @ ( lattic4823215512031491691ax_num @ M8 ) @ ( lattic4823215512031491691ax_num @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6923_Max__mono,axiom,
    ! [M8: set_nat,N7: set_nat] :
      ( ( ord_less_eq_set_nat @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N7 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M8 ) @ ( lattic8265883725875713057ax_nat @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6924_Max__mono,axiom,
    ! [M8: set_int,N7: set_int] :
      ( ( ord_less_eq_set_int @ M8 @ N7 )
     => ( ( M8 != bot_bot_set_int )
       => ( ( finite_finite_int @ N7 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M8 ) @ ( lattic8263393255366662781ax_int @ N7 ) ) ) ) ) ).

% Max_mono
thf(fact_6925_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_6926_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_6927_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_6928_negD,axiom,
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ zero_zero_int )
     => ? [N2: nat] :
          ( X4
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_6929_foldl__set,axiom,
    ! [L: list_set_real] :
      ( ( foldl_4142838678319929319t_real @ sup_sup_set_real @ bot_bot_set_real @ L )
      = ( comple3096694443085538997t_real
        @ ( collect_set_real
          @ ^ [X: set_real] : ( member_set_real @ X @ ( set_set_real2 @ L ) ) ) ) ) ).

% foldl_set
thf(fact_6930_foldl__set,axiom,
    ! [L: list_set_o] :
      ( ( foldl_set_o_set_o @ sup_sup_set_o @ bot_bot_set_o @ L )
      = ( comple90263536869209701_set_o
        @ ( collect_set_o
          @ ^ [X: set_o] : ( member_set_o @ X @ ( set_set_o2 @ L ) ) ) ) ) ).

% foldl_set
thf(fact_6931_foldl__set,axiom,
    ! [L: list_set_int] :
      ( ( foldl_6819690284573351271et_int @ sup_sup_set_int @ bot_bot_set_int @ L )
      = ( comple3221217463730067765et_int
        @ ( collect_set_int
          @ ^ [X: set_int] : ( member_set_int @ X @ ( set_set_int2 @ L ) ) ) ) ) ).

% foldl_set
thf(fact_6932_foldl__set,axiom,
    ! [L: list_set_nat] :
      ( ( foldl_4988731653086973103et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ L )
      = ( comple7399068483239264473et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] : ( member_set_nat @ X @ ( set_set_nat2 @ L ) ) ) ) ) ).

% foldl_set
thf(fact_6933_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_6934_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A2: real] : ( if_real @ ( A2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_6935_Max__add__commute,axiom,
    ! [S3: set_complex,F: complex > real,K: real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic4275903605611617917x_real
            @ ( image_complex_real
              @ ^ [X: complex] : ( plus_plus_real @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_complex_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6936_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > real,K: real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic4275903605611617917x_real
            @ ( image_7738145705984076560r_real
              @ ^ [X: code_integer] : ( plus_plus_real @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_7738145705984076560r_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6937_Max__add__commute,axiom,
    ! [S3: set_complex,F: complex > rat,K: rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_complex_rat
              @ ^ [X: complex] : ( plus_plus_rat @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_complex_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6938_Max__add__commute,axiom,
    ! [S3: set_Code_integer,F: code_integer > rat,K: rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_315895873841295420er_rat
              @ ^ [X: code_integer] : ( plus_plus_rat @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_315895873841295420er_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6939_Max__add__commute,axiom,
    ! [S3: set_real,F: real > real,K: real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( lattic4275903605611617917x_real
            @ ( image_real_real
              @ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_real_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6940_Max__add__commute,axiom,
    ! [S3: set_real,F: real > rat,K: rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_real_rat
              @ ^ [X: real] : ( plus_plus_rat @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_real_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6941_Max__add__commute,axiom,
    ! [S3: set_o,F: $o > real,K: real] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ( ( lattic4275903605611617917x_real
            @ ( image_o_real
              @ ^ [X: $o] : ( plus_plus_real @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_o_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6942_Max__add__commute,axiom,
    ! [S3: set_o,F: $o > rat,K: rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ( ( lattic7630753665789217321ax_rat
            @ ( image_o_rat
              @ ^ [X: $o] : ( plus_plus_rat @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_o_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6943_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
              @ ^ [X: nat] : ( plus_plus_real @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_real @ ( lattic4275903605611617917x_real @ ( image_nat_real @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6944_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
              @ ^ [X: nat] : ( plus_plus_rat @ ( F @ X ) @ K )
              @ S3 ) )
          = ( plus_plus_rat @ ( lattic7630753665789217321ax_rat @ ( image_nat_rat @ F @ S3 ) ) @ K ) ) ) ) ).

% Max_add_commute
thf(fact_6945_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A5: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat )
            @ A5 ) ) ) ) ).

% Id_on_def
thf(fact_6946_Id__on__def,axiom,
    ( id_on_int
    = ( ^ [A5: set_int] :
          ( comple5382143125604098054nt_int
          @ ( image_8635204845542730022nt_int
            @ ^ [X: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int )
            @ A5 ) ) ) ) ).

% Id_on_def
thf(fact_6947_Id__on__def,axiom,
    ( id_on_Code_integer
    = ( ^ [A5: set_Code_integer] :
          ( comple2203973573673791208nteger
          @ ( image_8529064383210301591nteger
            @ ^ [X: code_integer] : ( insert4913895101485356395nteger @ ( produc1086072967326762835nteger @ X @ X ) @ bot_bo4276436098303576167nteger )
            @ A5 ) ) ) ) ).

% Id_on_def
thf(fact_6948_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_6949_zminus1__lemma,axiom,
    ! [A: int,B: int,Q5: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R3 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q5 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q5 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_6950_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_6951_UN__insert,axiom,
    ! [B4: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_6952_UN__insert,axiom,
    ! [B4: vEBT_VEBT > set_nat,A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ B4 @ ( insert_VEBT_VEBT @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ B4 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_6953_UN__insert,axiom,
    ! [B4: int > set_nat,A: int,A3: set_int] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ ( insert_int @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_6954_UN__insert,axiom,
    ! [B4: $o > set_nat,A: $o,A3: set_o] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_6955_UN__insert,axiom,
    ! [B4: real > set_nat,A: real,A3: set_real] :
      ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ ( insert_real @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_6956_UN__simps_I2_J,axiom,
    ! [C2: set_real,A3: real > set_real,B4: set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6957_UN__simps_I2_J,axiom,
    ! [C2: set_real,A3: real > set_o,B4: set_o] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_real_set_o @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6958_UN__simps_I2_J,axiom,
    ! [C2: set_real,A3: real > set_int,B4: set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6959_UN__simps_I2_J,axiom,
    ! [C2: set_o,A3: $o > set_real,B4: set_real] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6960_UN__simps_I2_J,axiom,
    ! [C2: set_o,A3: $o > set_o,B4: set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6961_UN__simps_I2_J,axiom,
    ! [C2: set_o,A3: $o > set_int,B4: set_int] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6962_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A3: nat > set_real,B4: set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6963_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A3: nat > set_o,B4: set_o] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X: nat] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X: nat] : ( sup_sup_set_o @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6964_UN__simps_I2_J,axiom,
    ! [C2: set_nat,A3: nat > set_int,B4: set_int] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [X: nat] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [X: nat] : ( sup_sup_set_int @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6965_UN__simps_I2_J,axiom,
    ! [C2: set_int,A3: int > set_real,B4: set_real] :
      ( ( ( C2 = bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [X: int] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [X: int] : ( sup_sup_set_real @ ( A3 @ X ) @ B4 )
              @ C2 ) )
          = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_int_set_real @ A3 @ C2 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_6966_UN__simps_I3_J,axiom,
    ! [C2: set_real,A3: set_real,B4: real > set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_real @ A3 @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6967_UN__simps_I3_J,axiom,
    ! [C2: set_real,A3: set_o,B4: real > set_o] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6968_UN__simps_I3_J,axiom,
    ! [C2: set_real,A3: set_int,B4: real > set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_int @ A3 @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6969_UN__simps_I3_J,axiom,
    ! [C2: set_o,A3: set_real,B4: $o > set_real] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_real @ A3 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6970_UN__simps_I3_J,axiom,
    ! [C2: set_o,A3: set_o,B4: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6971_UN__simps_I3_J,axiom,
    ! [C2: set_o,A3: set_int,B4: $o > set_int] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_int @ A3 @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6972_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A3: set_real,B4: nat > set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_real @ A3 @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6973_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A3: set_o,B4: nat > set_o] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X: nat] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X: nat] : ( sup_sup_set_o @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6974_UN__simps_I3_J,axiom,
    ! [C2: set_nat,A3: set_int,B4: nat > set_int] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [X: nat] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [X: nat] : ( sup_sup_set_int @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_int @ A3 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6975_UN__simps_I3_J,axiom,
    ! [C2: set_int,A3: set_real,B4: int > set_real] :
      ( ( ( C2 = bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [X: int] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [X: int] : ( sup_sup_set_real @ A3 @ ( B4 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_real @ A3 @ ( comple3096694443085538997t_real @ ( image_int_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_6976_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: vEBT_VEBT,B4: real > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_6925917818215209377T_VEBT
              @ ^ [X: real] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_6925917818215209377T_VEBT
              @ ^ [X: real] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6977_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: real,B4: real > set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6978_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: $o,B4: real > set_o] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6979_UN__simps_I1_J,axiom,
    ! [C2: set_real,A: int,B4: real > set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6980_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: vEBT_VEBT,B4: $o > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_7704241249472752129T_VEBT
              @ ^ [X: $o] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_7704241249472752129T_VEBT
              @ ^ [X: $o] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_7704241249472752129T_VEBT @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6981_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: real,B4: $o > set_real] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6982_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: $o,B4: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6983_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: int,B4: $o > set_int] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_int ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6984_UN__simps_I1_J,axiom,
    ! [C2: set_nat,A: vEBT_VEBT,B4: nat > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_1406951880692228733T_VEBT
              @ ^ [X: nat] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple2820511241208326657T_VEBT
            @ ( image_1406951880692228733T_VEBT
              @ ^ [X: nat] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6985_UN__simps_I1_J,axiom,
    ! [C2: set_nat,A: real,B4: nat > set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = bot_bot_set_real ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) )
          = ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_6986_UN__singleton,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( comple2820511241208326657T_VEBT
        @ ( image_2685870239581809509T_VEBT
          @ ^ [X: vEBT_VEBT] : ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_6987_UN__singleton,axiom,
    ! [A3: set_real] :
      ( ( comple3096694443085538997t_real
        @ ( image_real_set_real
          @ ^ [X: real] : ( insert_real @ X @ bot_bot_set_real )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_6988_UN__singleton,axiom,
    ! [A3: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_6989_UN__singleton,axiom,
    ! [A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [X: int] : ( insert_int @ X @ bot_bot_set_int )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_6990_UN__singleton,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( insert_nat @ X @ bot_bot_set_nat )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_6991_ccSUP__empty,axiom,
    ! [F: real > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ bot_bot_set_real ) )
      = bot_bot_set_real ) ).

% ccSUP_empty
thf(fact_6992_ccSUP__empty,axiom,
    ! [F: real > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_real_set_o @ F @ bot_bot_set_real ) )
      = bot_bot_set_o ) ).

% ccSUP_empty
thf(fact_6993_ccSUP__empty,axiom,
    ! [F: real > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ bot_bot_set_real ) )
      = bot_bot_set_int ) ).

% ccSUP_empty
thf(fact_6994_ccSUP__empty,axiom,
    ! [F: $o > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ bot_bot_set_o ) )
      = bot_bot_set_real ) ).

% ccSUP_empty
thf(fact_6995_ccSUP__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% ccSUP_empty
thf(fact_6996_ccSUP__empty,axiom,
    ! [F: $o > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_o_set_int @ F @ bot_bot_set_o ) )
      = bot_bot_set_int ) ).

% ccSUP_empty
thf(fact_6997_ccSUP__empty,axiom,
    ! [F: nat > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ bot_bot_set_nat ) )
      = bot_bot_set_real ) ).

% ccSUP_empty
thf(fact_6998_ccSUP__empty,axiom,
    ! [F: nat > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ bot_bot_set_nat ) )
      = bot_bot_set_o ) ).

% ccSUP_empty
thf(fact_6999_ccSUP__empty,axiom,
    ! [F: nat > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ bot_bot_set_nat ) )
      = bot_bot_set_int ) ).

% ccSUP_empty
thf(fact_7000_ccSUP__empty,axiom,
    ! [F: int > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_int_set_real @ F @ bot_bot_set_int ) )
      = bot_bot_set_real ) ).

% ccSUP_empty
thf(fact_7001_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_real] :
      ( ( ( comple3096694443085538997t_real @ A3 )
        = bot_bot_set_real )
      = ( ! [X: set_real] :
            ( ( member_set_real @ X @ A3 )
           => ( X = bot_bot_set_real ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_7002_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A3 )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A3 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_7003_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_int] :
      ( ( ( comple3221217463730067765et_int @ A3 )
        = bot_bot_set_int )
      = ( ! [X: set_int] :
            ( ( member_set_int @ X @ A3 )
           => ( X = bot_bot_set_int ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_7004_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_7005_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_real] :
      ( ( bot_bot_set_real
        = ( comple3096694443085538997t_real @ A3 ) )
      = ( ! [X: set_real] :
            ( ( member_set_real @ X @ A3 )
           => ( X = bot_bot_set_real ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_7006_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A3 ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A3 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_7007_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ A3 ) )
      = ( ! [X: set_int] :
            ( ( member_set_int @ X @ A3 )
           => ( X = bot_bot_set_int ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_7008_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_7009_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_7010_SUP__identity__eq,axiom,
    ! [A3: set_int] :
      ( ( complete_Sup_Sup_int
        @ ( image_int_int
          @ ^ [X: int] : X
          @ A3 ) )
      = ( complete_Sup_Sup_int @ A3 ) ) ).

% SUP_identity_eq
thf(fact_7011_SUP__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Sup_Sup_nat
        @ ( image_nat_nat
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( complete_Sup_Sup_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_7012_SUP__identity__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [X: set_nat] : X
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_7013_SUP__identity__eq,axiom,
    ! [A3: set_real] :
      ( ( comple1385675409528146559p_real
        @ ( image_real_real
          @ ^ [X: real] : X
          @ A3 ) )
      = ( comple1385675409528146559p_real @ A3 ) ) ).

% SUP_identity_eq
thf(fact_7014_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: vEBT_VEBT,B4: nat > set_VEBT_VEBT] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_VEBT_VEBT @ B @ ( B4 @ A ) )
       => ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7015_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: real,B4: nat > set_real] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_real @ B @ ( B4 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7016_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: int,B4: nat > set_int] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_int @ B @ ( B4 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7017_UN__I,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B: vEBT_VEBT,B4: vEBT_VEBT > set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ( member_VEBT_VEBT @ B @ ( B4 @ A ) )
       => ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_2685870239581809509T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7018_UN__I,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B: real,B4: vEBT_VEBT > set_real] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ( member_real @ B @ ( B4 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_6636839513470643793t_real @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7019_UN__I,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B: int,B4: vEBT_VEBT > set_int] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ( member_int @ B @ ( B4 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7020_UN__I,axiom,
    ! [A: real,A3: set_real,B: vEBT_VEBT,B4: real > set_VEBT_VEBT] :
      ( ( member_real @ A @ A3 )
     => ( ( member_VEBT_VEBT @ B @ ( B4 @ A ) )
       => ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7021_UN__I,axiom,
    ! [A: real,A3: set_real,B: real,B4: real > set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( member_real @ B @ ( B4 @ A ) )
       => ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7022_UN__I,axiom,
    ! [A: real,A3: set_real,B: int,B4: real > set_int] :
      ( ( member_real @ A @ A3 )
     => ( ( member_int @ B @ ( B4 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7023_UN__I,axiom,
    ! [A: int,A3: set_int,B: vEBT_VEBT,B4: int > set_VEBT_VEBT] :
      ( ( member_int @ A @ A3 )
     => ( ( member_VEBT_VEBT @ B @ ( B4 @ A ) )
       => ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_1216245965459964705T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_7024_UN__iff,axiom,
    ! [B: nat,B4: nat > set_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( member_nat @ B @ ( B4 @ X ) ) ) ) ) ).

% UN_iff
thf(fact_7025_Sup__empty,axiom,
    ( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
    = bot_bot_set_real ) ).

% Sup_empty
thf(fact_7026_Sup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Sup_empty
thf(fact_7027_Sup__empty,axiom,
    ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
    = bot_bot_set_int ) ).

% Sup_empty
thf(fact_7028_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_7029_Sup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Sup_empty
thf(fact_7030_ccSup__empty,axiom,
    ( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
    = bot_bot_set_real ) ).

% ccSup_empty
thf(fact_7031_ccSup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% ccSup_empty
thf(fact_7032_ccSup__empty,axiom,
    ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
    = bot_bot_set_int ) ).

% ccSup_empty
thf(fact_7033_ccSup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% ccSup_empty
thf(fact_7034_ccSup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% ccSup_empty
thf(fact_7035_SUP__bot__conv_I2_J,axiom,
    ! [B4: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_7036_SUP__bot__conv_I1_J,axiom,
    ! [B4: nat > set_nat,A3: set_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_7037_SUP__bot,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% SUP_bot
thf(fact_7038_ccSUP__bot,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% ccSUP_bot
thf(fact_7039_SUP__const,axiom,
    ! [A3: set_real,F: set_nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ( comple7399068483239264473et_nat
          @ ( image_real_set_nat
            @ ^ [I3: real] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_7040_SUP__const,axiom,
    ! [A3: set_o,F: set_nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple7399068483239264473et_nat
          @ ( image_o_set_nat
            @ ^ [I3: $o] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_7041_SUP__const,axiom,
    ! [A3: set_nat,F: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [I3: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_7042_SUP__const,axiom,
    ! [A3: set_int,F: set_nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple7399068483239264473et_nat
          @ ( image_int_set_nat
            @ ^ [I3: int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_7043_ccSUP__const,axiom,
    ! [A3: set_real,F: set_nat] :
      ( ( A3 != bot_bot_set_real )
     => ( ( comple7399068483239264473et_nat
          @ ( image_real_set_nat
            @ ^ [I3: real] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_7044_ccSUP__const,axiom,
    ! [A3: set_o,F: set_nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple7399068483239264473et_nat
          @ ( image_o_set_nat
            @ ^ [I3: $o] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_7045_ccSUP__const,axiom,
    ! [A3: set_nat,F: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [I3: nat] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_7046_ccSUP__const,axiom,
    ! [A3: set_int,F: set_nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple7399068483239264473et_nat
          @ ( image_int_set_nat
            @ ^ [I3: int] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_7047_UN__constant,axiom,
    ! [A3: set_real,C: set_real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7048_UN__constant,axiom,
    ! [A3: set_real,C: set_o] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7049_UN__constant,axiom,
    ! [A3: set_real,C: set_int] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7050_UN__constant,axiom,
    ! [A3: set_o,C: set_real] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7051_UN__constant,axiom,
    ! [A3: set_o,C: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7052_UN__constant,axiom,
    ! [A3: set_o,C: set_int] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7053_UN__constant,axiom,
    ! [A3: set_nat,C: set_real] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7054_UN__constant,axiom,
    ! [A3: set_nat,C: set_o] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7055_UN__constant,axiom,
    ! [A3: set_nat,C: set_int] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7056_UN__constant,axiom,
    ! [A3: set_int,C: set_real] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [Y4: int] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [Y4: int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_7057_UN__Un,axiom,
    ! [M8: nat > set_nat,A3: set_nat,B4: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ ( sup_sup_set_nat @ A3 @ B4 ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ B4 ) ) ) ) ).

% UN_Un
thf(fact_7058_Sup__set__def,axiom,
    ( comple2820511241208326657T_VEBT
    = ( ^ [A5: set_set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] : ( complete_Sup_Sup_o @ ( image_5801891848100486793VEBT_o @ ( member_VEBT_VEBT @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7059_Sup__set__def,axiom,
    ( comple3096694443085538997t_real
    = ( ^ [A5: set_set_real] :
          ( collect_real
          @ ^ [X: real] : ( complete_Sup_Sup_o @ ( image_set_real_o @ ( member_real @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7060_Sup__set__def,axiom,
    ( comple8424636186594484919omplex
    = ( ^ [A5: set_set_complex] :
          ( collect_complex
          @ ^ [X: complex] : ( complete_Sup_Sup_o @ ( image_set_complex_o @ ( member_complex @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7061_Sup__set__def,axiom,
    ( comple8404747032580312297st_nat
    = ( ^ [A5: set_set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] : ( complete_Sup_Sup_o @ ( image_set_list_nat_o @ ( member_list_nat @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7062_Sup__set__def,axiom,
    ( comple548664676211718543et_nat
    = ( ^ [A5: set_set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] : ( complete_Sup_Sup_o @ ( image_set_set_nat_o @ ( member_set_nat @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7063_Sup__set__def,axiom,
    ( comple3221217463730067765et_int
    = ( ^ [A5: set_set_int] :
          ( collect_int
          @ ^ [X: int] : ( complete_Sup_Sup_o @ ( image_set_int_o @ ( member_int @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7064_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_7065_SUP__Sup__eq,axiom,
    ! [S3: set_set_VEBT_VEBT] :
      ( ( comple3152016690827447324VEBT_o
        @ ( image_7157618159374383128VEBT_o
          @ ^ [I3: set_VEBT_VEBT,X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ I3 )
          @ S3 ) )
      = ( ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( comple2820511241208326657T_VEBT @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_7066_SUP__Sup__eq,axiom,
    ! [S3: set_set_real] :
      ( ( comple3015195443809154064real_o
        @ ( image_5650221686686655994real_o
          @ ^ [I3: set_real,X: real] : ( member_real @ X @ I3 )
          @ S3 ) )
      = ( ^ [X: real] : ( member_real @ X @ ( comple3096694443085538997t_real @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_7067_SUP__Sup__eq,axiom,
    ! [S3: set_set_int] :
      ( ( comple6496622788309502864_int_o
        @ ( image_set_int_int_o
          @ ^ [I3: set_int,X: int] : ( member_int @ X @ I3 )
          @ S3 ) )
      = ( ^ [X: int] : ( member_int @ X @ ( comple3221217463730067765et_int @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_7068_SUP__Sup__eq,axiom,
    ! [S3: set_set_set_nat] :
      ( ( comple3806919086088850358_nat_o
        @ ( image_4331731847045299910_nat_o
          @ ^ [I3: set_set_nat,X: set_nat] : ( member_set_nat @ X @ I3 )
          @ S3 ) )
      = ( ^ [X: set_nat] : ( member_set_nat @ X @ ( comple548664676211718543et_nat @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_7069_SUP__Sup__eq,axiom,
    ! [S3: set_set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I3: set_nat,X: nat] : ( member_nat @ X @ I3 )
          @ S3 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_7070_SUP__UN__eq,axiom,
    ! [R3: nat > set_nat,S3: set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_nat_nat_o
          @ ^ [I3: nat,X: nat] : ( member_nat @ X @ ( R3 @ I3 ) )
          @ S3 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R3 @ S3 ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_7071_SUP__Sup__eq2,axiom,
    ! [S3: set_se7855581050983116737at_nat] :
      ( ( comple3592611370556534995_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I3: set_Pr1261947904930325089at_nat,X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ I3 )
          @ S3 ) )
      = ( ^ [X: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( comple5685304695842803022at_nat @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_7072_SUP__Sup__eq2,axiom,
    ! [S3: set_se3932177096832370463BT_nat] :
      ( ( comple5511761119775097859_nat_o
        @ ( image_4158226296049136288_nat_o
          @ ^ [I3: set_Pr7556676689462069481BT_nat,X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ I3 )
          @ S3 ) )
      = ( ^ [X: vEBT_VEBT,Y4: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y4 ) @ ( comple9061401370350521660BT_nat @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_7073_SUP__Sup__eq2,axiom,
    ! [S3: set_se6260736226359567993nt_int] :
      ( ( comple7687260386943045147_int_o
        @ ( image_4446226961036766042_int_o
          @ ^ [I3: set_Pr958786334691620121nt_int,X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ I3 )
          @ S3 ) )
      = ( ^ [X: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ ( comple5382143125604098054nt_int @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_7074_SUP__Sup__eq2,axiom,
    ! [S3: set_se4344029326803248219nteger] :
      ( ( comple6471460710946744121eger_o
        @ ( image_2931711902234744598eger_o
          @ ^ [I3: set_Pr4811707699266497531nteger,X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ I3 )
          @ S3 ) )
      = ( ^ [X: code_integer,Y4: code_integer] : ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y4 ) @ ( comple2203973573673791208nteger @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_7075_SUP__Sup__eq2,axiom,
    ! [S3: set_se4826145725398303499at_num] :
      ( ( comple4350791933526045961_num_o
        @ ( image_2055017250723459638_num_o
          @ ^ [I3: set_Pr6200539531224447659at_num,X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ I3 )
          @ S3 ) )
      = ( ^ [X: nat,Y4: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y4 ) @ ( comple1400524285282149784at_num @ S3 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_7076_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_7077_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_nat > nat,A3: set_nat] :
      ( ( Inf
        @ ( image_nat_nat
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_7078_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_int > int,A3: set_int] :
      ( ( Inf
        @ ( image_int_int
          @ ^ [X: int] : X
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_7079_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_nat > nat,A3: set_nat] :
      ( ( Sup
        @ ( image_nat_nat
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_7080_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_int > int,A3: set_int] :
      ( ( Sup
        @ ( image_int_int
          @ ^ [X: int] : X
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_7081_Sup__upper2,axiom,
    ! [U: set_int,A3: set_set_int,V: set_int] :
      ( ( member_set_int @ U @ A3 )
     => ( ( ord_less_eq_set_int @ V @ U )
       => ( ord_less_eq_set_int @ V @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).

% Sup_upper2
thf(fact_7082_Sup__upper2,axiom,
    ! [U: set_nat,A3: set_set_nat,V: set_nat] :
      ( ( member_set_nat @ U @ A3 )
     => ( ( ord_less_eq_set_nat @ V @ U )
       => ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% Sup_upper2
thf(fact_7083_Sup__le__iff,axiom,
    ! [A3: set_set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ B )
      = ( ! [X: set_int] :
            ( ( member_set_int @ X @ A3 )
           => ( ord_less_eq_set_int @ X @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_7084_Sup__le__iff,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ B )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( ord_less_eq_set_nat @ X @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_7085_Sup__upper,axiom,
    ! [X4: set_int,A3: set_set_int] :
      ( ( member_set_int @ X4 @ A3 )
     => ( ord_less_eq_set_int @ X4 @ ( comple3221217463730067765et_int @ A3 ) ) ) ).

% Sup_upper
thf(fact_7086_Sup__upper,axiom,
    ! [X4: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X4 @ A3 )
     => ( ord_less_eq_set_nat @ X4 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ).

% Sup_upper
thf(fact_7087_Sup__least,axiom,
    ! [A3: set_set_int,Z: set_int] :
      ( ! [X3: set_int] :
          ( ( member_set_int @ X3 @ A3 )
         => ( ord_less_eq_set_int @ X3 @ Z ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_7088_Sup__least,axiom,
    ! [A3: set_set_nat,Z: set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ X3 @ Z ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_7089_Sup__mono,axiom,
    ! [A3: set_set_int,B4: set_set_int] :
      ( ! [A4: set_int] :
          ( ( member_set_int @ A4 @ A3 )
         => ? [X6: set_int] :
              ( ( member_set_int @ X6 @ B4 )
              & ( ord_less_eq_set_int @ A4 @ X6 ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B4 ) ) ) ).

% Sup_mono
thf(fact_7090_Sup__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ! [A4: set_nat] :
          ( ( member_set_nat @ A4 @ A3 )
         => ? [X6: set_nat] :
              ( ( member_set_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ A4 @ X6 ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).

% Sup_mono
thf(fact_7091_Sup__eqI,axiom,
    ! [A3: set_set_int,X4: set_int] :
      ( ! [Y3: set_int] :
          ( ( member_set_int @ Y3 @ A3 )
         => ( ord_less_eq_set_int @ Y3 @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [Z5: set_int] :
                ( ( member_set_int @ Z5 @ A3 )
               => ( ord_less_eq_set_int @ Z5 @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ A3 )
          = X4 ) ) ) ).

% Sup_eqI
thf(fact_7092_Sup__eqI,axiom,
    ! [A3: set_set_nat,X4: set_nat] :
      ( ! [Y3: set_nat] :
          ( ( member_set_nat @ Y3 @ A3 )
         => ( ord_less_eq_set_nat @ Y3 @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [Z5: set_nat] :
                ( ( member_set_nat @ Z5 @ A3 )
               => ( ord_less_eq_set_nat @ Z5 @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ A3 )
          = X4 ) ) ) ).

% Sup_eqI
thf(fact_7093_empty__Union__conv,axiom,
    ! [A3: set_set_real] :
      ( ( bot_bot_set_real
        = ( comple3096694443085538997t_real @ A3 ) )
      = ( ! [X: set_real] :
            ( ( member_set_real @ X @ A3 )
           => ( X = bot_bot_set_real ) ) ) ) ).

% empty_Union_conv
thf(fact_7094_empty__Union__conv,axiom,
    ! [A3: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A3 ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A3 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% empty_Union_conv
thf(fact_7095_empty__Union__conv,axiom,
    ! [A3: set_set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ A3 ) )
      = ( ! [X: set_int] :
            ( ( member_set_int @ X @ A3 )
           => ( X = bot_bot_set_int ) ) ) ) ).

% empty_Union_conv
thf(fact_7096_empty__Union__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = bot_bot_set_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_7097_Union__empty__conv,axiom,
    ! [A3: set_set_real] :
      ( ( ( comple3096694443085538997t_real @ A3 )
        = bot_bot_set_real )
      = ( ! [X: set_real] :
            ( ( member_set_real @ X @ A3 )
           => ( X = bot_bot_set_real ) ) ) ) ).

% Union_empty_conv
thf(fact_7098_Union__empty__conv,axiom,
    ! [A3: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A3 )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A3 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Union_empty_conv
thf(fact_7099_Union__empty__conv,axiom,
    ! [A3: set_set_int] :
      ( ( ( comple3221217463730067765et_int @ A3 )
        = bot_bot_set_int )
      = ( ! [X: set_int] :
            ( ( member_set_int @ X @ A3 )
           => ( X = bot_bot_set_int ) ) ) ) ).

% Union_empty_conv
thf(fact_7100_Union__empty__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ( X = bot_bot_set_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_7101_Union__empty,axiom,
    ( ( comple3096694443085538997t_real @ bot_bot_set_set_real )
    = bot_bot_set_real ) ).

% Union_empty
thf(fact_7102_Union__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Union_empty
thf(fact_7103_Union__empty,axiom,
    ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
    = bot_bot_set_int ) ).

% Union_empty
thf(fact_7104_Union__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Union_empty
thf(fact_7105_Union__subsetI,axiom,
    ! [A3: set_set_int,B4: set_set_int] :
      ( ! [X3: set_int] :
          ( ( member_set_int @ X3 @ A3 )
         => ? [Y5: set_int] :
              ( ( member_set_int @ Y5 @ B4 )
              & ( ord_less_eq_set_int @ X3 @ Y5 ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B4 ) ) ) ).

% Union_subsetI
thf(fact_7106_Union__subsetI,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ? [Y5: set_nat] :
              ( ( member_set_nat @ Y5 @ B4 )
              & ( ord_less_eq_set_nat @ X3 @ Y5 ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).

% Union_subsetI
thf(fact_7107_Union__upper,axiom,
    ! [B4: set_int,A3: set_set_int] :
      ( ( member_set_int @ B4 @ A3 )
     => ( ord_less_eq_set_int @ B4 @ ( comple3221217463730067765et_int @ A3 ) ) ) ).

% Union_upper
thf(fact_7108_Union__upper,axiom,
    ! [B4: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ B4 @ A3 )
     => ( ord_less_eq_set_nat @ B4 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ).

% Union_upper
thf(fact_7109_Union__least,axiom,
    ! [A3: set_set_int,C2: set_int] :
      ( ! [X10: set_int] :
          ( ( member_set_int @ X10 @ A3 )
         => ( ord_less_eq_set_int @ X10 @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ C2 ) ) ).

% Union_least
thf(fact_7110_Union__least,axiom,
    ! [A3: set_set_nat,C2: set_nat] :
      ( ! [X10: set_nat] :
          ( ( member_set_nat @ X10 @ A3 )
         => ( ord_less_eq_set_nat @ X10 @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ C2 ) ) ).

% Union_least
thf(fact_7111_Union__mono,axiom,
    ! [A3: set_set_int,B4: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B4 ) ) ) ).

% Union_mono
thf(fact_7112_Union__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).

% Union_mono
thf(fact_7113_SUP__commute,axiom,
    ! [F: nat > nat > set_nat,B4: set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I3 ) @ B4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [J3: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [I3: nat] : ( F @ I3 @ J3 )
                @ A3 ) )
          @ B4 ) ) ) ).

% SUP_commute
thf(fact_7114_UN__extend__simps_I9_J,axiom,
    ! [C2: nat > set_nat,B4: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B4 @ X ) ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_7115_UN__E,axiom,
    ! [B: vEBT_VEBT,B4: nat > set_VEBT_VEBT,A3: set_nat] :
      ( ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ A3 ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ~ ( member_VEBT_VEBT @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7116_UN__E,axiom,
    ! [B: vEBT_VEBT,B4: vEBT_VEBT > set_VEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_2685870239581809509T_VEBT @ B4 @ A3 ) ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ~ ( member_VEBT_VEBT @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7117_UN__E,axiom,
    ! [B: vEBT_VEBT,B4: real > set_VEBT_VEBT,A3: set_real] :
      ( ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member_VEBT_VEBT @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7118_UN__E,axiom,
    ! [B: vEBT_VEBT,B4: int > set_VEBT_VEBT,A3: set_int] :
      ( ( member_VEBT_VEBT @ B @ ( comple2820511241208326657T_VEBT @ ( image_1216245965459964705T_VEBT @ B4 @ A3 ) ) )
     => ~ ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ~ ( member_VEBT_VEBT @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7119_UN__E,axiom,
    ! [B: real,B4: nat > set_real,A3: set_nat] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ A3 ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7120_UN__E,axiom,
    ! [B: real,B4: vEBT_VEBT > set_real,A3: set_VEBT_VEBT] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_6636839513470643793t_real @ B4 @ A3 ) ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7121_UN__E,axiom,
    ! [B: real,B4: real > set_real,A3: set_real] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ A3 ) ) )
     => ~ ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7122_UN__E,axiom,
    ! [B: real,B4: int > set_real,A3: set_int] :
      ( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_int_set_real @ B4 @ A3 ) ) )
     => ~ ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ~ ( member_real @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7123_UN__E,axiom,
    ! [B: int,B4: nat > set_int,A3: set_nat] :
      ( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ~ ( member_int @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7124_UN__E,axiom,
    ! [B: int,B4: vEBT_VEBT > set_int,A3: set_VEBT_VEBT] :
      ( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) )
     => ~ ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ~ ( member_int @ B @ ( B4 @ X3 ) ) ) ) ).

% UN_E
thf(fact_7125_UN__UN__flatten,axiom,
    ! [C2: nat > set_nat,B4: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B4 @ Y4 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_7126_SUP__eq,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_int,G: nat > set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B4 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7127_SUP__eq,axiom,
    ! [A3: set_nat,B4: set_VEBT_VEBT,F: nat > set_int,G: vEBT_VEBT > set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ? [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ J2 @ B4 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7128_SUP__eq,axiom,
    ! [A3: set_nat,B4: set_real,F: nat > set_int,G: real > set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ? [X6: real] :
              ( ( member_real @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: real] :
            ( ( member_real @ J2 @ B4 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_real_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7129_SUP__eq,axiom,
    ! [A3: set_nat,B4: set_int,F: nat > set_int,G: int > set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ? [X6: int] :
              ( ( member_int @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: int] :
            ( ( member_int @ J2 @ B4 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7130_SUP__eq,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_nat,F: vEBT_VEBT > set_int,G: nat > set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B4 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7131_SUP__eq,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,F: vEBT_VEBT > set_int,G: vEBT_VEBT > set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ? [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ J2 @ B4 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7132_SUP__eq,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_real,F: vEBT_VEBT > set_int,G: real > set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ? [X6: real] :
              ( ( member_real @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: real] :
            ( ( member_real @ J2 @ B4 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_real_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7133_SUP__eq,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_int,F: vEBT_VEBT > set_int,G: int > set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ? [X6: int] :
              ( ( member_int @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: int] :
            ( ( member_int @ J2 @ B4 )
           => ? [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7134_SUP__eq,axiom,
    ! [A3: set_real,B4: set_nat,F: real > set_int,G: nat > set_int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B4 )
           => ? [X6: real] :
                ( ( member_real @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7135_SUP__eq,axiom,
    ! [A3: set_real,B4: set_VEBT_VEBT,F: real > set_int,G: vEBT_VEBT > set_int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ? [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ B4 )
              & ( ord_less_eq_set_int @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
     => ( ! [J2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ J2 @ B4 )
           => ? [X6: real] :
                ( ( member_real @ X6 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ G @ B4 ) ) ) ) ) ).

% SUP_eq
thf(fact_7136_less__eq__Sup,axiom,
    ! [A3: set_o,U: $o] :
      ( ! [V2: $o] :
          ( ( member_o @ V2 @ A3 )
         => ( ord_less_eq_o @ U @ V2 ) )
     => ( ( A3 != bot_bot_set_o )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ).

% less_eq_Sup
thf(fact_7137_less__eq__Sup,axiom,
    ! [A3: set_set_int,U: set_int] :
      ( ! [V2: set_int] :
          ( ( member_set_int @ V2 @ A3 )
         => ( ord_less_eq_set_int @ U @ V2 ) )
     => ( ( A3 != bot_bot_set_set_int )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).

% less_eq_Sup
thf(fact_7138_less__eq__Sup,axiom,
    ! [A3: set_set_nat,U: set_nat] :
      ( ! [V2: set_nat] :
          ( ( member_set_nat @ V2 @ A3 )
         => ( ord_less_eq_set_nat @ U @ V2 ) )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% less_eq_Sup
thf(fact_7139_Sup__subset__mono,axiom,
    ! [A3: set_set_int,B4: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B4 )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B4 ) ) ) ).

% Sup_subset_mono
thf(fact_7140_Sup__subset__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).

% Sup_subset_mono
thf(fact_7141_SUP__eq__const,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,X4: set_nat] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7142_SUP__eq__const,axiom,
    ! [I5: set_set_nat,F: set_nat > set_nat,X4: set_nat] :
      ( ( I5 != bot_bot_set_set_nat )
     => ( ! [I2: set_nat] :
            ( ( member_set_nat @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7143_SUP__eq__const,axiom,
    ! [I5: set_real,F: real > set_nat,X4: set_nat] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7144_SUP__eq__const,axiom,
    ! [I5: set_o,F: $o > set_nat,X4: set_nat] :
      ( ( I5 != bot_bot_set_o )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7145_SUP__eq__const,axiom,
    ! [I5: set_nat,F: nat > set_nat,X4: set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7146_SUP__eq__const,axiom,
    ! [I5: set_int,F: int > set_nat,X4: set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ( F @ I2 )
              = X4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I5 ) )
          = X4 ) ) ) ).

% SUP_eq_const
thf(fact_7147_SUP__eqI,axiom,
    ! [A3: set_nat,F: nat > set_int,X4: set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7148_SUP__eqI,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > set_int,X4: set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7149_SUP__eqI,axiom,
    ! [A3: set_real,F: real > set_int,X4: set_int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [I4: real] :
                ( ( member_real @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7150_SUP__eqI,axiom,
    ! [A3: set_int,F: int > set_int,X4: set_int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [I4: int] :
                ( ( member_int @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7151_SUP__eqI,axiom,
    ! [A3: set_set_nat,F: set_nat > set_int,X4: set_int] :
      ( ! [I2: set_nat] :
          ( ( member_set_nat @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_int] :
            ( ! [I4: set_nat] :
                ( ( member_set_nat @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_int @ X4 @ Y3 ) )
       => ( ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7152_SUP__eqI,axiom,
    ! [A3: set_nat,F: nat > set_nat,X4: set_nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7153_SUP__eqI,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,X4: set_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7154_SUP__eqI,axiom,
    ! [A3: set_real,F: real > set_nat,X4: set_nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: real] :
                ( ( member_real @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7155_SUP__eqI,axiom,
    ! [A3: set_int,F: int > set_nat,X4: set_nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: int] :
                ( ( member_int @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7156_SUP__eqI,axiom,
    ! [A3: set_set_nat,F: set_nat > set_nat,X4: set_nat] :
      ( ! [I2: set_nat] :
          ( ( member_set_nat @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ X4 ) )
     => ( ! [Y3: set_nat] :
            ( ! [I4: set_nat] :
                ( ( member_set_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
           => ( ord_less_eq_set_nat @ X4 @ Y3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) )
          = X4 ) ) ) ).

% SUP_eqI
thf(fact_7157_SUP__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ).

% SUP_mono
thf(fact_7158_SUP__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_nat,F: vEBT_VEBT > set_nat,G: nat > set_nat] :
      ( ! [N2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ N2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ).

% SUP_mono
thf(fact_7159_SUP__mono,axiom,
    ! [A3: set_real,B4: set_nat,F: real > set_nat,G: nat > set_nat] :
      ( ! [N2: real] :
          ( ( member_real @ N2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ).

% SUP_mono
thf(fact_7160_SUP__mono,axiom,
    ! [A3: set_int,B4: set_nat,F: int > set_nat,G: nat > set_nat] :
      ( ! [N2: int] :
          ( ( member_int @ N2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ).

% SUP_mono
thf(fact_7161_SUP__mono,axiom,
    ! [A3: set_set_nat,B4: set_nat,F: set_nat > set_nat,G: nat > set_nat] :
      ( ! [N2: set_nat] :
          ( ( member_set_nat @ N2 @ A3 )
         => ? [X6: nat] :
              ( ( member_nat @ X6 @ B4 )
              & ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ).

% SUP_mono
thf(fact_7162_SUP__least,axiom,
    ! [A3: set_nat,F: nat > set_int,U: set_int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7163_SUP__least,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > set_int,U: set_int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7164_SUP__least,axiom,
    ! [A3: set_real,F: real > set_int,U: set_int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7165_SUP__least,axiom,
    ! [A3: set_int,F: int > set_int,U: set_int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7166_SUP__least,axiom,
    ! [A3: set_set_nat,F: set_nat > set_int,U: set_int] :
      ( ! [I2: set_nat] :
          ( ( member_set_nat @ I2 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7167_SUP__least,axiom,
    ! [A3: set_nat,F: nat > set_nat,U: set_nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7168_SUP__least,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,U: set_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7169_SUP__least,axiom,
    ! [A3: set_real,F: real > set_nat,U: set_nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7170_SUP__least,axiom,
    ! [A3: set_int,F: int > set_nat,U: set_nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7171_SUP__least,axiom,
    ! [A3: set_set_nat,F: set_nat > set_nat,U: set_nat] :
      ( ! [I2: set_nat] :
          ( ( member_set_nat @ I2 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_7172_SUP__mono_H,axiom,
    ! [F: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ! [X3: nat] : ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_7173_SUP__upper,axiom,
    ! [I: nat,A3: set_nat,F: nat > set_int] :
      ( ( member_nat @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7174_SUP__upper,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > set_int] :
      ( ( member_VEBT_VEBT @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7175_SUP__upper,axiom,
    ! [I: real,A3: set_real,F: real > set_int] :
      ( ( member_real @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7176_SUP__upper,axiom,
    ! [I: int,A3: set_int,F: int > set_int] :
      ( ( member_int @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7177_SUP__upper,axiom,
    ! [I: set_nat,A3: set_set_nat,F: set_nat > set_int] :
      ( ( member_set_nat @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7178_SUP__upper,axiom,
    ! [I: nat,A3: set_nat,F: nat > set_nat] :
      ( ( member_nat @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7179_SUP__upper,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > set_nat] :
      ( ( member_VEBT_VEBT @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7180_SUP__upper,axiom,
    ! [I: real,A3: set_real,F: real > set_nat] :
      ( ( member_real @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7181_SUP__upper,axiom,
    ! [I: int,A3: set_int,F: int > set_nat] :
      ( ( member_int @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7182_SUP__upper,axiom,
    ! [I: set_nat,A3: set_set_nat,F: set_nat > set_nat] :
      ( ( member_set_nat @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_7183_SUP__le__iff,axiom,
    ! [F: nat > set_nat,A3: set_nat,U: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ U )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_7184_SUP__upper2,axiom,
    ! [I: nat,A3: set_nat,U: set_int,F: nat > set_int] :
      ( ( member_nat @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7185_SUP__upper2,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,U: set_int,F: vEBT_VEBT > set_int] :
      ( ( member_VEBT_VEBT @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7186_SUP__upper2,axiom,
    ! [I: real,A3: set_real,U: set_int,F: real > set_int] :
      ( ( member_real @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7187_SUP__upper2,axiom,
    ! [I: int,A3: set_int,U: set_int,F: int > set_int] :
      ( ( member_int @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7188_SUP__upper2,axiom,
    ! [I: set_nat,A3: set_set_nat,U: set_int,F: set_nat > set_int] :
      ( ( member_set_nat @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7189_SUP__upper2,axiom,
    ! [I: nat,A3: set_nat,U: set_nat,F: nat > set_nat] :
      ( ( member_nat @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7190_SUP__upper2,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,U: set_nat,F: vEBT_VEBT > set_nat] :
      ( ( member_VEBT_VEBT @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7191_SUP__upper2,axiom,
    ! [I: real,A3: set_real,U: set_nat,F: real > set_nat] :
      ( ( member_real @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7192_SUP__upper2,axiom,
    ! [I: int,A3: set_int,U: set_nat,F: int > set_nat] :
      ( ( member_int @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7193_SUP__upper2,axiom,
    ! [I: set_nat,A3: set_set_nat,U: set_nat,F: set_nat > set_nat] :
      ( ( member_set_nat @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_7194_SUP__lessD,axiom,
    ! [F: nat > set_nat,A3: set_nat,Y: set_nat,I: nat] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ Y )
     => ( ( member_nat @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_7195_SUP__lessD,axiom,
    ! [F: vEBT_VEBT > set_nat,A3: set_VEBT_VEBT,Y: set_nat,I: vEBT_VEBT] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) @ Y )
     => ( ( member_VEBT_VEBT @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_7196_SUP__lessD,axiom,
    ! [F: real > set_nat,A3: set_real,Y: set_nat,I: real] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) @ Y )
     => ( ( member_real @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_7197_SUP__lessD,axiom,
    ! [F: int > set_nat,A3: set_int,Y: set_nat,I: int] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ Y )
     => ( ( member_int @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_7198_SUP__lessD,axiom,
    ! [F: set_nat > set_nat,A3: set_set_nat,Y: set_nat,I: set_nat] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) @ Y )
     => ( ( member_set_nat @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_7199_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [F: nat > set_nat,A3: set_nat,G: nat > set_nat] :
      ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( sup_sup_set_nat @ ( F @ A2 ) @ ( G @ A2 ) )
          @ A3 ) ) ) ).

% complete_lattice_class.SUP_sup_distrib
thf(fact_7200_SUP__absorb,axiom,
    ! [K: nat,I5: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) ) ) ).

% SUP_absorb
thf(fact_7201_SUP__absorb,axiom,
    ! [K: vEBT_VEBT,I5: set_VEBT_VEBT,A3: vEBT_VEBT > set_nat] :
      ( ( member_VEBT_VEBT @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ A3 @ I5 ) ) ) ) ).

% SUP_absorb
thf(fact_7202_SUP__absorb,axiom,
    ! [K: real,I5: set_real,A3: real > set_nat] :
      ( ( member_real @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ I5 ) ) ) ) ).

% SUP_absorb
thf(fact_7203_SUP__absorb,axiom,
    ! [K: int,I5: set_int,A3: int > set_nat] :
      ( ( member_int @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ I5 ) ) ) ) ).

% SUP_absorb
thf(fact_7204_SUP__absorb,axiom,
    ! [K: set_nat,I5: set_set_nat,A3: set_nat > set_nat] :
      ( ( member_set_nat @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A3 @ I5 ) ) ) ) ).

% SUP_absorb
thf(fact_7205_UN__extend__simps_I10_J,axiom,
    ! [B4: int > set_nat,F: int > int,A3: set_int] :
      ( ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [A2: int] : ( B4 @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ ( image_int_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_7206_UN__extend__simps_I10_J,axiom,
    ! [B4: real > set_nat,F: nat > real,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B4 @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ ( image_nat_real @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_7207_UN__extend__simps_I10_J,axiom,
    ! [B4: set_nat > set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B4 @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B4 @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_7208_UN__extend__simps_I10_J,axiom,
    ! [B4: int > set_nat,F: nat > int,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B4 @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ ( image_nat_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_7209_UN__extend__simps_I10_J,axiom,
    ! [B4: nat > set_nat,F: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B4 @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_7210_image__UN,axiom,
    ! [F: nat > real,B4: nat > set_nat,A3: set_nat] :
      ( ( image_nat_real @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( comple3096694443085538997t_real
        @ ( image_nat_set_real
          @ ^ [X: nat] : ( image_nat_real @ F @ ( B4 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_7211_image__UN,axiom,
    ! [F: nat > set_nat,B4: nat > set_nat,A3: set_nat] :
      ( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( image_nat_set_nat @ F @ ( B4 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_7212_image__UN,axiom,
    ! [F: nat > int,B4: nat > set_nat,A3: set_nat] :
      ( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [X: nat] : ( image_nat_int @ F @ ( B4 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_7213_image__UN,axiom,
    ! [F: nat > nat,B4: nat > set_nat,A3: set_nat] :
      ( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( image_nat_nat @ F @ ( B4 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_7214_UN__empty2,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% UN_empty2
thf(fact_7215_UN__empty,axiom,
    ! [B4: real > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ bot_bot_set_real ) )
      = bot_bot_set_real ) ).

% UN_empty
thf(fact_7216_UN__empty,axiom,
    ! [B4: real > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_real_set_o @ B4 @ bot_bot_set_real ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_7217_UN__empty,axiom,
    ! [B4: real > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ bot_bot_set_real ) )
      = bot_bot_set_int ) ).

% UN_empty
thf(fact_7218_UN__empty,axiom,
    ! [B4: $o > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_o_set_real @ B4 @ bot_bot_set_o ) )
      = bot_bot_set_real ) ).

% UN_empty
thf(fact_7219_UN__empty,axiom,
    ! [B4: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_7220_UN__empty,axiom,
    ! [B4: $o > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_o_set_int @ B4 @ bot_bot_set_o ) )
      = bot_bot_set_int ) ).

% UN_empty
thf(fact_7221_UN__empty,axiom,
    ! [B4: nat > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ bot_bot_set_nat ) )
      = bot_bot_set_real ) ).

% UN_empty
thf(fact_7222_UN__empty,axiom,
    ! [B4: nat > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ B4 @ bot_bot_set_nat ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_7223_UN__empty,axiom,
    ! [B4: nat > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ bot_bot_set_nat ) )
      = bot_bot_set_int ) ).

% UN_empty
thf(fact_7224_UN__empty,axiom,
    ! [B4: int > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_int_set_real @ B4 @ bot_bot_set_int ) )
      = bot_bot_set_real ) ).

% UN_empty
thf(fact_7225_UNION__empty__conv_I1_J,axiom,
    ! [B4: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_7226_UNION__empty__conv_I2_J,axiom,
    ! [B4: nat > set_nat,A3: set_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_7227_UN__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_int,G: nat > set_int] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7228_UN__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,F: vEBT_VEBT > set_int,G: vEBT_VEBT > set_int] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7229_UN__mono,axiom,
    ! [A3: set_real,B4: set_real,F: real > set_int,G: real > set_int] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_real_set_int @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7230_UN__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,F: set_nat > set_int,G: set_nat > set_int] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7231_UN__mono,axiom,
    ! [A3: set_int,B4: set_int,F: int > set_int,G: int > set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7232_UN__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7233_UN__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,G: vEBT_VEBT > set_nat] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7234_UN__mono,axiom,
    ! [A3: set_real,B4: set_real,F: real > set_nat,G: real > set_nat] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7235_UN__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7236_UN__mono,axiom,
    ! [A3: set_int,B4: set_int,F: int > set_nat,G: int > set_nat] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B4 ) ) ) ) ) ).

% UN_mono
thf(fact_7237_UN__least,axiom,
    ! [A3: set_nat,B4: nat > set_int,C2: set_int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_set_int @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7238_UN__least,axiom,
    ! [A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_int,C2: set_int] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_set_int @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7239_UN__least,axiom,
    ! [A3: set_real,B4: real > set_int,C2: set_int] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_set_int @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7240_UN__least,axiom,
    ! [A3: set_int,B4: int > set_int,C2: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_set_int @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7241_UN__least,axiom,
    ! [A3: set_set_nat,B4: set_nat > set_int,C2: set_int] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( ord_less_eq_set_int @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7242_UN__least,axiom,
    ! [A3: set_nat,B4: nat > set_nat,C2: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7243_UN__least,axiom,
    ! [A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_nat,C2: set_nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7244_UN__least,axiom,
    ! [A3: set_real,B4: real > set_nat,C2: set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7245_UN__least,axiom,
    ! [A3: set_int,B4: int > set_nat,C2: set_nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7246_UN__least,axiom,
    ! [A3: set_set_nat,B4: set_nat > set_nat,C2: set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B4 @ A3 ) ) @ C2 ) ) ).

% UN_least
thf(fact_7247_UN__upper,axiom,
    ! [A: nat,A3: set_nat,B4: nat > set_int] :
      ( ( member_nat @ A @ A3 )
     => ( ord_less_eq_set_int @ ( B4 @ A ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7248_UN__upper,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_int] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ord_less_eq_set_int @ ( B4 @ A ) @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7249_UN__upper,axiom,
    ! [A: real,A3: set_real,B4: real > set_int] :
      ( ( member_real @ A @ A3 )
     => ( ord_less_eq_set_int @ ( B4 @ A ) @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7250_UN__upper,axiom,
    ! [A: int,A3: set_int,B4: int > set_int] :
      ( ( member_int @ A @ A3 )
     => ( ord_less_eq_set_int @ ( B4 @ A ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7251_UN__upper,axiom,
    ! [A: set_nat,A3: set_set_nat,B4: set_nat > set_int] :
      ( ( member_set_nat @ A @ A3 )
     => ( ord_less_eq_set_int @ ( B4 @ A ) @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7252_UN__upper,axiom,
    ! [A: nat,A3: set_nat,B4: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7253_UN__upper,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,B4: vEBT_VEBT > set_nat] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7254_UN__upper,axiom,
    ! [A: real,A3: set_real,B4: real > set_nat] :
      ( ( member_real @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7255_UN__upper,axiom,
    ! [A: int,A3: set_int,B4: int > set_nat] :
      ( ( member_int @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7256_UN__upper,axiom,
    ! [A: set_nat,A3: set_set_nat,B4: set_nat > set_nat] :
      ( ( member_set_nat @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B4 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B4 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_7257_UN__subset__iff,axiom,
    ! [A3: nat > set_nat,I5: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I5 )
           => ( ord_less_eq_set_nat @ ( A3 @ X ) @ B4 ) ) ) ) ).

% UN_subset_iff
thf(fact_7258_UN__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: vEBT_VEBT,B4: nat > set_VEBT_VEBT] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple2820511241208326657T_VEBT
          @ ( image_1406951880692228733T_VEBT
            @ ^ [X: nat] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7259_UN__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: int,B4: nat > set_int] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple3221217463730067765et_int
          @ ( image_nat_set_int
            @ ^ [X: nat] : ( insert_int @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7260_UN__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: $o,B4: nat > set_o] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_nat_set_o
            @ ^ [X: nat] : ( insert_o @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7261_UN__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: real,B4: nat > set_real] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple3096694443085538997t_real
          @ ( image_nat_set_real
            @ ^ [X: nat] : ( insert_real @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7262_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A3: set_VEBT_VEBT,A: vEBT_VEBT,B4: vEBT_VEBT > set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ U @ A3 )
     => ( ( comple2820511241208326657T_VEBT
          @ ( image_2685870239581809509T_VEBT
            @ ^ [X: vEBT_VEBT] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_2685870239581809509T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7263_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A3: set_VEBT_VEBT,A: int,B4: vEBT_VEBT > set_int] :
      ( ( member_VEBT_VEBT @ U @ A3 )
     => ( ( comple3221217463730067765et_int
          @ ( image_2273570491937255121et_int
            @ ^ [X: vEBT_VEBT] : ( insert_int @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7264_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A3: set_VEBT_VEBT,A: $o,B4: vEBT_VEBT > set_o] :
      ( ( member_VEBT_VEBT @ U @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_7883550159813902793_set_o
            @ ^ [X: vEBT_VEBT] : ( insert_o @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_7883550159813902793_set_o @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7265_UN__insert__distrib,axiom,
    ! [U: vEBT_VEBT,A3: set_VEBT_VEBT,A: real,B4: vEBT_VEBT > set_real] :
      ( ( member_VEBT_VEBT @ U @ A3 )
     => ( ( comple3096694443085538997t_real
          @ ( image_6636839513470643793t_real
            @ ^ [X: vEBT_VEBT] : ( insert_real @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_6636839513470643793t_real @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7266_UN__insert__distrib,axiom,
    ! [U: real,A3: set_real,A: vEBT_VEBT,B4: real > set_VEBT_VEBT] :
      ( ( member_real @ U @ A3 )
     => ( ( comple2820511241208326657T_VEBT
          @ ( image_6925917818215209377T_VEBT
            @ ^ [X: real] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7267_UN__insert__distrib,axiom,
    ! [U: real,A3: set_real,A: int,B4: real > set_int] :
      ( ( member_real @ U @ A3 )
     => ( ( comple3221217463730067765et_int
          @ ( image_real_set_int
            @ ^ [X: real] : ( insert_int @ A @ ( B4 @ X ) )
            @ A3 ) )
        = ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_7268_UN__extend__simps_I6_J,axiom,
    ! [A3: nat > set_nat,C2: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C2 ) ) @ B4 )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( minus_minus_set_nat @ ( A3 @ X ) @ B4 )
          @ C2 ) ) ) ).

% UN_extend_simps(6)
thf(fact_7269_UN__absorb,axiom,
    ! [K: nat,I5: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) ) ) ).

% UN_absorb
thf(fact_7270_UN__absorb,axiom,
    ! [K: vEBT_VEBT,I5: set_VEBT_VEBT,A3: vEBT_VEBT > set_nat] :
      ( ( member_VEBT_VEBT @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ A3 @ I5 ) ) ) ) ).

% UN_absorb
thf(fact_7271_UN__absorb,axiom,
    ! [K: real,I5: set_real,A3: real > set_nat] :
      ( ( member_real @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ I5 ) ) ) ) ).

% UN_absorb
thf(fact_7272_UN__absorb,axiom,
    ! [K: int,I5: set_int,A3: int > set_nat] :
      ( ( member_int @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ I5 ) ) ) ) ).

% UN_absorb
thf(fact_7273_UN__absorb,axiom,
    ! [K: set_nat,I5: set_set_nat,A3: set_nat > set_nat] :
      ( ( member_set_nat @ K @ I5 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A3 @ I5 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A3 @ I5 ) ) ) ) ).

% UN_absorb
thf(fact_7274_UN__Un__distrib,axiom,
    ! [A3: nat > set_nat,B4: nat > set_nat,I5: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I3: nat] : ( sup_sup_set_nat @ ( A3 @ I3 ) @ ( B4 @ I3 ) )
          @ I5 ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I5 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ I5 ) ) ) ) ).

% UN_Un_distrib
thf(fact_7275_Un__Union__image,axiom,
    ! [A3: nat > set_nat,B4: nat > set_nat,C2: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ ( B4 @ X ) )
          @ C2 ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ C2 ) ) ) ) ).

% Un_Union_image
thf(fact_7276_image__Union,axiom,
    ! [F: int > int,S3: set_set_int] :
      ( ( image_int_int @ F @ ( comple3221217463730067765et_int @ S3 ) )
      = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_7277_image__Union,axiom,
    ! [F: nat > real,S3: set_set_nat] :
      ( ( image_nat_real @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple3096694443085538997t_real @ ( image_6333053925516494319t_real @ ( image_nat_real @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_7278_image__Union,axiom,
    ! [F: nat > set_nat,S3: set_set_nat] :
      ( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_7279_image__Union,axiom,
    ! [F: nat > int,S3: set_set_nat] :
      ( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_7280_image__Union,axiom,
    ! [F: nat > nat,S3: set_set_nat] :
      ( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_7281_UN__extend__simps_I8_J,axiom,
    ! [B4: nat > set_nat,A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [Y4: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ Y4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_7282_SUP__eq__iff,axiom,
    ! [I5: set_VEBT_VEBT,C: set_int,F: vEBT_VEBT > set_int] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I2 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ I5 ) )
            = C )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7283_SUP__eq__iff,axiom,
    ! [I5: set_real,C: set_int,F: real > set_int] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I2 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ I5 ) )
            = C )
          = ( ! [X: real] :
                ( ( member_real @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7284_SUP__eq__iff,axiom,
    ! [I5: set_o,C: set_int,F: $o > set_int] :
      ( ( I5 != bot_bot_set_o )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I2 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_o_set_int @ F @ I5 ) )
            = C )
          = ( ! [X: $o] :
                ( ( member_o @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7285_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_int,F: nat > set_int] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I2 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ I5 ) )
            = C )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7286_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_int,F: int > set_int] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I2 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ I5 ) )
            = C )
          = ( ! [X: int] :
                ( ( member_int @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7287_SUP__eq__iff,axiom,
    ! [I5: set_VEBT_VEBT,C: set_nat,F: vEBT_VEBT > set_nat] :
      ( ( I5 != bot_bo8194388402131092736T_VEBT )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I2 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ I5 ) )
            = C )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7288_SUP__eq__iff,axiom,
    ! [I5: set_real,C: set_nat,F: real > set_nat] :
      ( ( I5 != bot_bot_set_real )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I2 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X: real] :
                ( ( member_real @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7289_SUP__eq__iff,axiom,
    ! [I5: set_o,C: set_nat,F: $o > set_nat] :
      ( ( I5 != bot_bot_set_o )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I2 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X: $o] :
                ( ( member_o @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7290_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_nat,F: nat > set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I2 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7291_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_nat,F: int > set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I2 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X: int] :
                ( ( member_int @ X @ I5 )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_7292_SUP__subset__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_int,G: nat > set_int] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7293_SUP__subset__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,F: vEBT_VEBT > set_int,G: vEBT_VEBT > set_int] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7294_SUP__subset__mono,axiom,
    ! [A3: set_real,B4: set_real,F: real > set_int,G: real > set_int] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_real_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7295_SUP__subset__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,F: set_nat > set_int,G: set_nat > set_int] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7296_SUP__subset__mono,axiom,
    ! [A3: set_int,B4: set_int,F: int > set_int,G: int > set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7297_SUP__subset__mono,axiom,
    ! [A3: set_nat,B4: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7298_SUP__subset__mono,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,G: vEBT_VEBT > set_nat] :
      ( ( ord_le4337996190870823476T_VEBT @ A3 @ B4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7299_SUP__subset__mono,axiom,
    ! [A3: set_real,B4: set_real,F: real > set_nat,G: real > set_nat] :
      ( ( ord_less_eq_set_real @ A3 @ B4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7300_SUP__subset__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7301_SUP__subset__mono,axiom,
    ! [A3: set_int,B4: set_int,F: int > set_nat,G: int > set_nat] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B4 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_7302_SUP__constant,axiom,
    ! [A3: set_real,C: set_real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7303_SUP__constant,axiom,
    ! [A3: set_real,C: set_o] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7304_SUP__constant,axiom,
    ! [A3: set_real,C: set_int] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [Y4: real] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [Y4: real] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7305_SUP__constant,axiom,
    ! [A3: set_o,C: set_real] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7306_SUP__constant,axiom,
    ! [A3: set_o,C: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7307_SUP__constant,axiom,
    ! [A3: set_o,C: set_int] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [Y4: $o] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7308_SUP__constant,axiom,
    ! [A3: set_nat,C: set_real] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7309_SUP__constant,axiom,
    ! [A3: set_nat,C: set_o] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7310_SUP__constant,axiom,
    ! [A3: set_nat,C: set_int] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y4: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7311_SUP__constant,axiom,
    ! [A3: set_int,C: set_real] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [Y4: int] : C
              @ A3 ) )
          = bot_bot_set_real ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple3096694443085538997t_real
            @ ( image_int_set_real
              @ ^ [Y4: int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_7312_SUP__empty,axiom,
    ! [F: real > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ bot_bot_set_real ) )
      = bot_bot_set_real ) ).

% SUP_empty
thf(fact_7313_SUP__empty,axiom,
    ! [F: real > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_real_set_o @ F @ bot_bot_set_real ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_7314_SUP__empty,axiom,
    ! [F: real > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_real_set_int @ F @ bot_bot_set_real ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_7315_SUP__empty,axiom,
    ! [F: $o > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ bot_bot_set_o ) )
      = bot_bot_set_real ) ).

% SUP_empty
thf(fact_7316_SUP__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_7317_SUP__empty,axiom,
    ! [F: $o > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_o_set_int @ F @ bot_bot_set_o ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_7318_SUP__empty,axiom,
    ! [F: nat > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ bot_bot_set_nat ) )
      = bot_bot_set_real ) ).

% SUP_empty
thf(fact_7319_SUP__empty,axiom,
    ! [F: nat > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ bot_bot_set_nat ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_7320_SUP__empty,axiom,
    ! [F: nat > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ bot_bot_set_nat ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_7321_SUP__empty,axiom,
    ! [F: int > set_real] :
      ( ( comple3096694443085538997t_real @ ( image_int_set_real @ F @ bot_bot_set_int ) )
      = bot_bot_set_real ) ).

% SUP_empty
thf(fact_7322_SUP__insert,axiom,
    ! [F: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_7323_SUP__insert,axiom,
    ! [F: vEBT_VEBT > set_nat,A: vEBT_VEBT,A3: set_VEBT_VEBT] :
      ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ ( insert_VEBT_VEBT @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_7324_SUP__insert,axiom,
    ! [F: int > set_nat,A: int,A3: set_int] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( insert_int @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_7325_SUP__insert,axiom,
    ! [F: $o > set_nat,A: $o,A3: set_o] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_7326_SUP__insert,axiom,
    ! [F: real > set_nat,A: real,A3: set_real] :
      ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ ( insert_real @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_7327_SUP__union,axiom,
    ! [M8: nat > set_nat,A3: set_nat,B4: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ ( sup_sup_set_nat @ A3 @ B4 ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M8 @ B4 ) ) ) ) ).

% SUP_union
thf(fact_7328_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: vEBT_VEBT,B4: real > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ C2 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_6925917818215209377T_VEBT @ B4 @ C2 ) ) )
          = ( comple2820511241208326657T_VEBT
            @ ( image_6925917818215209377T_VEBT
              @ ^ [X: real] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7329_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: real,B4: real > set_real] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ C2 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B4 @ C2 ) ) )
          = ( comple3096694443085538997t_real
            @ ( image_real_set_real
              @ ^ [X: real] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7330_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: $o,B4: real > set_o] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B4 @ C2 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B4 @ C2 ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_real_set_o
              @ ^ [X: real] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7331_UN__extend__simps_I1_J,axiom,
    ! [C2: set_real,A: int,B4: real > set_int] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ C2 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_real_set_int @ B4 @ C2 ) ) )
          = ( comple3221217463730067765et_int
            @ ( image_real_set_int
              @ ^ [X: real] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7332_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: vEBT_VEBT,B4: $o > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_7704241249472752129T_VEBT @ B4 @ C2 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_7704241249472752129T_VEBT @ B4 @ C2 ) ) )
          = ( comple2820511241208326657T_VEBT
            @ ( image_7704241249472752129T_VEBT
              @ ^ [X: $o] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7333_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: real,B4: $o > set_real] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B4 @ C2 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B4 @ C2 ) ) )
          = ( comple3096694443085538997t_real
            @ ( image_o_set_real
              @ ^ [X: $o] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7334_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: $o,B4: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B4 @ C2 ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7335_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: int,B4: $o > set_int] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B4 @ C2 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_int @ A @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B4 @ C2 ) ) )
          = ( comple3221217463730067765et_int
            @ ( image_o_set_int
              @ ^ [X: $o] : ( insert_int @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7336_UN__extend__simps_I1_J,axiom,
    ! [C2: set_nat,A: vEBT_VEBT,B4: nat > set_VEBT_VEBT] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ C2 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( insert_VEBT_VEBT @ A @ ( comple2820511241208326657T_VEBT @ ( image_1406951880692228733T_VEBT @ B4 @ C2 ) ) )
          = ( comple2820511241208326657T_VEBT
            @ ( image_1406951880692228733T_VEBT
              @ ^ [X: nat] : ( insert_VEBT_VEBT @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7337_UN__extend__simps_I1_J,axiom,
    ! [C2: set_nat,A: real,B4: nat > set_real] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ C2 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( insert_real @ A @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B4 @ C2 ) ) )
          = ( comple3096694443085538997t_real
            @ ( image_nat_set_real
              @ ^ [X: nat] : ( insert_real @ A @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_7338_SUP__UNION,axiom,
    ! [F: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y4 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_7339_UN__extend__simps_I3_J,axiom,
    ! [C2: set_real,A3: set_nat,B4: real > set_nat] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ C2 ) ) )
          = A3 ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B4 @ C2 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_real_set_nat
              @ ^ [X: real] : ( sup_sup_set_nat @ A3 @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_7340_UN__extend__simps_I3_J,axiom,
    ! [C2: set_o,A3: set_nat,B4: $o > set_nat] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ C2 ) ) )
          = A3 ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ C2 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X: $o] : ( sup_sup_set_nat @ A3 @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_7341_UN__extend__simps_I3_J,axiom,
    ! [C2: set_nat,A3: set_nat,B4: nat > set_nat] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ C2 ) ) )
          = A3 ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ C2 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ A3 @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_7342_UN__extend__simps_I3_J,axiom,
    ! [C2: set_int,A3: set_nat,B4: int > set_nat] :
      ( ( ( C2 = bot_bot_set_int )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ C2 ) ) )
          = A3 ) )
      & ( ( C2 != bot_bot_set_int )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ C2 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [X: int] : ( sup_sup_set_nat @ A3 @ ( B4 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_7343_UN__extend__simps_I2_J,axiom,
    ! [C2: set_real,A3: real > set_nat,B4: set_nat] :
      ( ( ( C2 = bot_bot_set_real )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ C2 ) ) @ B4 )
          = B4 ) )
      & ( ( C2 != bot_bot_set_real )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ A3 @ C2 ) ) @ B4 )
          = ( comple7399068483239264473et_nat
            @ ( image_real_set_nat
              @ ^ [X: real] : ( sup_sup_set_nat @ ( A3 @ X ) @ B4 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_7344_UN__extend__simps_I2_J,axiom,
    ! [C2: set_o,A3: $o > set_nat,B4: set_nat] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ C2 ) ) @ B4 )
          = B4 ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ C2 ) ) @ B4 )
          = ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X: $o] : ( sup_sup_set_nat @ ( A3 @ X ) @ B4 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_7345_UN__extend__simps_I2_J,axiom,
    ! [C2: set_nat,A3: nat > set_nat,B4: set_nat] :
      ( ( ( C2 = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C2 ) ) @ B4 )
          = B4 ) )
      & ( ( C2 != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C2 ) ) @ B4 )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ B4 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_7346_UN__extend__simps_I2_J,axiom,
    ! [C2: set_int,A3: int > set_nat,B4: set_nat] :
      ( ( ( C2 = bot_bot_set_int )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ C2 ) ) @ B4 )
          = B4 ) )
      & ( ( C2 != bot_bot_set_int )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A3 @ C2 ) ) @ B4 )
          = ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [X: int] : ( sup_sup_set_nat @ ( A3 @ X ) @ B4 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_7347_UNION__singleton__eq__range,axiom,
    ! [F: nat > set_nat,A3: set_nat] :
      ( ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( insert_set_nat @ ( F @ X ) @ bot_bot_set_set_nat )
          @ A3 ) )
      = ( image_nat_set_nat @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_7348_UNION__singleton__eq__range,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ( comple3096694443085538997t_real
        @ ( image_nat_set_real
          @ ^ [X: nat] : ( insert_real @ ( F @ X ) @ bot_bot_set_real )
          @ A3 ) )
      = ( image_nat_real @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_7349_UNION__singleton__eq__range,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [X: nat] : ( insert_int @ ( F @ X ) @ bot_bot_set_int )
          @ A3 ) )
      = ( image_nat_int @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_7350_UNION__singleton__eq__range,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [X: int] : ( insert_int @ ( F @ X ) @ bot_bot_set_int )
          @ A3 ) )
      = ( image_int_int @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_7351_UNION__singleton__eq__range,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( insert_nat @ ( F @ X ) @ bot_bot_set_nat )
          @ A3 ) )
      = ( image_nat_nat @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_7352_UNION__fun__upd,axiom,
    ! [A3: vEBT_VEBT > set_real,I: vEBT_VEBT,B4: set_real,J4: set_VEBT_VEBT] :
      ( ( comple3096694443085538997t_real @ ( image_6636839513470643793t_real @ ( fun_up5367274378234922969t_real @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_6636839513470643793t_real @ A3 @ ( minus_5127226145743854075T_VEBT @ J4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) ) ) @ ( if_set_real @ ( member_VEBT_VEBT @ I @ J4 ) @ B4 @ bot_bot_set_real ) ) ) ).

% UNION_fun_upd
thf(fact_7353_UNION__fun__upd,axiom,
    ! [A3: vEBT_VEBT > set_o,I: vEBT_VEBT,B4: set_o,J4: set_VEBT_VEBT] :
      ( ( comple90263536869209701_set_o @ ( image_7883550159813902793_set_o @ ( fun_up8985983397501432385_set_o @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_7883550159813902793_set_o @ A3 @ ( minus_5127226145743854075T_VEBT @ J4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) ) ) @ ( if_set_o @ ( member_VEBT_VEBT @ I @ J4 ) @ B4 @ bot_bot_set_o ) ) ) ).

% UNION_fun_upd
thf(fact_7354_UNION__fun__upd,axiom,
    ! [A3: vEBT_VEBT > set_int,I: vEBT_VEBT,B4: set_int,J4: set_VEBT_VEBT] :
      ( ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ ( fun_up4996286513446587481et_int @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_2273570491937255121et_int @ A3 @ ( minus_5127226145743854075T_VEBT @ J4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) ) ) @ ( if_set_int @ ( member_VEBT_VEBT @ I @ J4 ) @ B4 @ bot_bot_set_int ) ) ) ).

% UNION_fun_upd
thf(fact_7355_UNION__fun__upd,axiom,
    ! [A3: real > set_real,I: real,B4: set_real,J4: set_real] :
      ( ( comple3096694443085538997t_real @ ( image_real_set_real @ ( fun_up567933838727694221t_real @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ A3 @ ( minus_minus_set_real @ J4 @ ( insert_real @ I @ bot_bot_set_real ) ) ) ) @ ( if_set_real @ ( member_real @ I @ J4 ) @ B4 @ bot_bot_set_real ) ) ) ).

% UNION_fun_upd
thf(fact_7356_UNION__fun__upd,axiom,
    ! [A3: real > set_o,I: real,B4: set_o,J4: set_real] :
      ( ( comple90263536869209701_set_o @ ( image_real_set_o @ ( fun_upd_real_set_o @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_real_set_o @ A3 @ ( minus_minus_set_real @ J4 @ ( insert_real @ I @ bot_bot_set_real ) ) ) ) @ ( if_set_o @ ( member_real @ I @ J4 ) @ B4 @ bot_bot_set_o ) ) ) ).

% UNION_fun_upd
thf(fact_7357_UNION__fun__upd,axiom,
    ! [A3: real > set_int,I: real,B4: set_int,J4: set_real] :
      ( ( comple3221217463730067765et_int @ ( image_real_set_int @ ( fun_upd_real_set_int @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_real_set_int @ A3 @ ( minus_minus_set_real @ J4 @ ( insert_real @ I @ bot_bot_set_real ) ) ) ) @ ( if_set_int @ ( member_real @ I @ J4 ) @ B4 @ bot_bot_set_int ) ) ) ).

% UNION_fun_upd
thf(fact_7358_UNION__fun__upd,axiom,
    ! [A3: $o > set_real,I: $o,B4: set_real,J4: set_o] :
      ( ( comple3096694443085538997t_real @ ( image_o_set_real @ ( fun_upd_o_set_real @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ A3 @ ( minus_minus_set_o @ J4 @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_real @ ( member_o @ I @ J4 ) @ B4 @ bot_bot_set_real ) ) ) ).

% UNION_fun_upd
thf(fact_7359_UNION__fun__upd,axiom,
    ! [A3: $o > set_o,I: $o,B4: set_o,J4: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A3 @ ( minus_minus_set_o @ J4 @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I @ J4 ) @ B4 @ bot_bot_set_o ) ) ) ).

% UNION_fun_upd
thf(fact_7360_UNION__fun__upd,axiom,
    ! [A3: $o > set_int,I: $o,B4: set_int,J4: set_o] :
      ( ( comple3221217463730067765et_int @ ( image_o_set_int @ ( fun_upd_o_set_int @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ A3 @ ( minus_minus_set_o @ J4 @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_int @ ( member_o @ I @ J4 ) @ B4 @ bot_bot_set_int ) ) ) ).

% UNION_fun_upd
thf(fact_7361_UNION__fun__upd,axiom,
    ! [A3: int > set_real,I: int,B4: set_real,J4: set_int] :
      ( ( comple3096694443085538997t_real @ ( image_int_set_real @ ( fun_upd_int_set_real @ A3 @ I @ B4 ) @ J4 ) )
      = ( sup_sup_set_real @ ( comple3096694443085538997t_real @ ( image_int_set_real @ A3 @ ( minus_minus_set_int @ J4 @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) @ ( if_set_real @ ( member_int @ I @ J4 ) @ B4 @ bot_bot_set_real ) ) ) ).

% UNION_fun_upd
thf(fact_7362_ccpo__Sup__singleton,axiom,
    ! [X4: $o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ X4 @ bot_bot_set_o ) )
      = X4 ) ).

% ccpo_Sup_singleton
thf(fact_7363_ccpo__Sup__singleton,axiom,
    ! [X4: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
      = X4 ) ).

% ccpo_Sup_singleton
thf(fact_7364_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_7365_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_7366_SUP__nat__binary,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ A3
        @ ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : B4
            @ ( collect_nat @ ( ord_less_nat @ zero_zero_nat ) ) ) ) )
      = ( sup_sup_set_nat @ A3 @ B4 ) ) ).

% SUP_nat_binary
thf(fact_7367_Union__image__empty,axiom,
    ! [A3: set_nat,F: real > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ bot_bot_set_real ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_7368_Union__image__empty,axiom,
    ! [A3: set_nat,F: $o > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_7369_Union__image__empty,axiom,
    ! [A3: set_nat,F: nat > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_7370_Union__image__empty,axiom,
    ! [A3: set_nat,F: int > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ bot_bot_set_int ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_7371_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7372_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7373_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7374_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7375_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_7376_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7377_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7378_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7379_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7380_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_7381_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
      = ( ( ord_le2932123472753598470d_enat @ B @ A )
        & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_7382_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_7383_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_7384_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_7385_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_7386_max__less__iff__conj,axiom,
    ! [X4: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X4 @ Y ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X4 @ Z )
        & ( ord_le72135733267957522d_enat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7387_max__less__iff__conj,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X4 @ Y ) @ Z )
      = ( ( ord_less_real @ X4 @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7388_max__less__iff__conj,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X4 @ Y ) @ Z )
      = ( ( ord_less_rat @ X4 @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7389_max__less__iff__conj,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X4 @ Y ) @ Z )
      = ( ( ord_less_num @ X4 @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7390_max__less__iff__conj,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X4 @ Y ) @ Z )
      = ( ( ord_less_nat @ X4 @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7391_max__less__iff__conj,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X4 @ Y ) @ Z )
      = ( ( ord_less_int @ X4 @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_7392_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7393_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7394_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7395_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7396_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7397_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_7398_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7399_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7400_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7401_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7402_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7403_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_7404_max__bot2,axiom,
    ! [X4: set_real] :
      ( ( ord_max_set_real @ X4 @ bot_bot_set_real )
      = X4 ) ).

% max_bot2
thf(fact_7405_max__bot2,axiom,
    ! [X4: set_o] :
      ( ( ord_max_set_o @ X4 @ bot_bot_set_o )
      = X4 ) ).

% max_bot2
thf(fact_7406_max__bot2,axiom,
    ! [X4: set_nat] :
      ( ( ord_max_set_nat @ X4 @ bot_bot_set_nat )
      = X4 ) ).

% max_bot2
thf(fact_7407_max__bot2,axiom,
    ! [X4: set_int] :
      ( ( ord_max_set_int @ X4 @ bot_bot_set_int )
      = X4 ) ).

% max_bot2
thf(fact_7408_max__bot2,axiom,
    ! [X4: nat] :
      ( ( ord_max_nat @ X4 @ bot_bot_nat )
      = X4 ) ).

% max_bot2
thf(fact_7409_max__bot2,axiom,
    ! [X4: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ X4 @ bot_bo4199563552545308370d_enat )
      = X4 ) ).

% max_bot2
thf(fact_7410_max__bot,axiom,
    ! [X4: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7411_max__bot,axiom,
    ! [X4: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7412_max__bot,axiom,
    ! [X4: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7413_max__bot,axiom,
    ! [X4: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7414_max__bot,axiom,
    ! [X4: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7415_max__bot,axiom,
    ! [X4: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X4 )
      = X4 ) ).

% max_bot
thf(fact_7416_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_7417_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_7418_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_7419_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_7420_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_7421_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_7422_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_7423_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_7424_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_7425_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_7426_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_7427_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_7428_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_7429_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_7430_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_7431_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_7432_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_7433_Max__insert,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X4 @ A3 ) )
          = ( ord_max_Code_integer @ X4 @ ( lattic4901227151466704046nteger @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7434_Max__insert,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( lattic4275903605611617917x_real @ ( insert_real @ X4 @ A3 ) )
          = ( ord_max_real @ X4 @ ( lattic4275903605611617917x_real @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7435_Max__insert,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X4 @ A3 ) )
          = ( ord_max_o @ X4 @ ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7436_Max__insert,axiom,
    ! [A3: set_Extended_enat,X4: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( lattic921264341876707157d_enat @ ( insert_Extended_enat @ X4 @ A3 ) )
          = ( ord_ma741700101516333627d_enat @ X4 @ ( lattic921264341876707157d_enat @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7437_Max__insert,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X4 @ A3 ) )
          = ( ord_max_nat @ X4 @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7438_Max__insert,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X4 @ A3 ) )
          = ( ord_max_int @ X4 @ ( lattic8263393255366662781ax_int @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_7439_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_7440_of__nat__max,axiom,
    ! [X4: nat,Y: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X4 @ Y ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X4 ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).

% of_nat_max
thf(fact_7441_of__nat__max,axiom,
    ! [X4: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X4 @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_7442_of__nat__max,axiom,
    ! [X4: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X4 @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_7443_of__nat__max,axiom,
    ! [X4: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X4 @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_7444_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7445_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7446_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7447_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7448_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_7449_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7450_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7451_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7452_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7453_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_7454_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7455_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7456_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7457_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7458_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_7459_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7460_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7461_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7462_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7463_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_7464_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X4: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X4 @ Y ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X4 )
        | ( ord_le2932123472753598470d_enat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7465_le__max__iff__disj,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X4 @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X4 )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7466_le__max__iff__disj,axiom,
    ! [Z: num,X4: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X4 @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X4 )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7467_le__max__iff__disj,axiom,
    ! [Z: nat,X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X4 @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X4 )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7468_le__max__iff__disj,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X4 @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X4 )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_7469_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7470_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7471_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_7472_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_7473_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_7474_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7475_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7476_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_7477_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_7478_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_7479_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( A2
          = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7480_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7481_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( A2
          = ( ord_max_num @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7482_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7483_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_7484_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ A )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7485_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7486_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7487_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7488_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_7489_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7490_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7491_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7492_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7493_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_7494_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_7495_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_7496_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_7497_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_7498_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_7499_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7500_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7501_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_7502_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_7503_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_7504_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ( ord_le2932123472753598470d_enat @ D @ B )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7505_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7506_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7507_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7508_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_7509_max__absorb2,axiom,
    ! [X4: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X4 @ Y )
     => ( ( ord_ma741700101516333627d_enat @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7510_max__absorb2,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_max_set_int @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7511_max__absorb2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_max_rat @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7512_max__absorb2,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_max_num @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7513_max__absorb2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_max_nat @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7514_max__absorb2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_max_int @ X4 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_7515_max__absorb1,axiom,
    ! [Y: extended_enat,X4: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X4 )
     => ( ( ord_ma741700101516333627d_enat @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7516_max__absorb1,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ( ( ord_max_set_int @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7517_max__absorb1,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ( ord_max_rat @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7518_max__absorb1,axiom,
    ! [Y: num,X4: num] :
      ( ( ord_less_eq_num @ Y @ X4 )
     => ( ( ord_max_num @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7519_max__absorb1,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ( ( ord_max_nat @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7520_max__absorb1,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ( ( ord_max_int @ X4 @ Y )
        = X4 ) ) ).

% max_absorb1
thf(fact_7521_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7522_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A2: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7523_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7524_max__def,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7525_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7526_max__def,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_7527_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X4: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X4 @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X4 )
        | ( ord_le72135733267957522d_enat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7528_less__max__iff__disj,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X4 @ Y ) )
      = ( ( ord_less_real @ Z @ X4 )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7529_less__max__iff__disj,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X4 @ Y ) )
      = ( ( ord_less_rat @ Z @ X4 )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7530_less__max__iff__disj,axiom,
    ! [Z: num,X4: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X4 @ Y ) )
      = ( ( ord_less_num @ Z @ X4 )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7531_less__max__iff__disj,axiom,
    ! [Z: nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X4 @ Y ) )
      = ( ( ord_less_nat @ Z @ X4 )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7532_less__max__iff__disj,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X4 @ Y ) )
      = ( ( ord_less_int @ Z @ X4 )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_7533_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7534_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7535_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7536_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7537_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7538_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_7539_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( A2
            = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7540_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( A2
            = ( ord_max_real @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7541_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( ord_max_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7542_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( A2
            = ( ord_max_num @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7543_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7544_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_7545_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7546_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7547_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7548_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7549_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7550_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_7551_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7552_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7553_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7554_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7555_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7556_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_7557_max__add__distrib__left,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X4 @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X4 @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7558_max__add__distrib__left,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X4 @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X4 @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7559_max__add__distrib__left,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X4 @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X4 @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7560_max__add__distrib__left,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X4 @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X4 @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_7561_max__add__distrib__right,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X4 @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X4 @ Y ) @ ( plus_plus_real @ X4 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7562_max__add__distrib__right,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X4 @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( plus_plus_rat @ X4 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7563_max__add__distrib__right,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X4 @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X4 @ Y ) @ ( plus_plus_nat @ X4 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7564_max__add__distrib__right,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X4 @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X4 @ Y ) @ ( plus_plus_int @ X4 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_7565_max__diff__distrib__left,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X4 @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X4 @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7566_max__diff__distrib__left,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X4 @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X4 @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7567_max__diff__distrib__left,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X4 @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X4 @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_7568_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q5 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q5 ) @ ( plus_plus_nat @ N @ Q5 ) ) ) ).

% nat_add_max_left
thf(fact_7569_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q5 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q5 ) ) ) ).

% nat_add_max_right
thf(fact_7570_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q5 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q5 ) @ ( times_times_nat @ N @ Q5 ) ) ) ).

% nat_mult_max_left
thf(fact_7571_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q5 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q5 ) ) ) ).

% nat_mult_max_right
thf(fact_7572_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7573_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A2: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7574_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7575_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7576_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7577_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_7578_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_7579_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X )
    @ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_7580_Sup__insert__finite,axiom,
    ! [S3: set_Extended_enat,X4: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( S3 = bot_bo7653980558646680370d_enat )
         => ( ( comple4398354569131411667d_enat @ ( insert_Extended_enat @ X4 @ S3 ) )
            = X4 ) )
        & ( ( S3 != bot_bo7653980558646680370d_enat )
         => ( ( comple4398354569131411667d_enat @ ( insert_Extended_enat @ X4 @ S3 ) )
            = ( ord_ma741700101516333627d_enat @ X4 @ ( comple4398354569131411667d_enat @ S3 ) ) ) ) ) ) ).

% Sup_insert_finite
thf(fact_7581_Sup__insert__finite,axiom,
    ! [S3: set_int,X4: int] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( S3 = bot_bot_set_int )
         => ( ( complete_Sup_Sup_int @ ( insert_int @ X4 @ S3 ) )
            = X4 ) )
        & ( ( S3 != bot_bot_set_int )
         => ( ( complete_Sup_Sup_int @ ( insert_int @ X4 @ S3 ) )
            = ( ord_max_int @ X4 @ ( complete_Sup_Sup_int @ S3 ) ) ) ) ) ) ).

% Sup_insert_finite
thf(fact_7582_Sup__insert__finite,axiom,
    ! [S3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( S3 = bot_bot_set_nat )
         => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X4 @ S3 ) )
            = X4 ) )
        & ( ( S3 != bot_bot_set_nat )
         => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X4 @ S3 ) )
            = ( ord_max_nat @ X4 @ ( complete_Sup_Sup_nat @ S3 ) ) ) ) ) ) ).

% Sup_insert_finite
thf(fact_7583_Sup__insert__finite,axiom,
    ! [S3: set_real,X4: real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( S3 = bot_bot_set_real )
         => ( ( comple1385675409528146559p_real @ ( insert_real @ X4 @ S3 ) )
            = X4 ) )
        & ( ( S3 != bot_bot_set_real )
         => ( ( comple1385675409528146559p_real @ ( insert_real @ X4 @ S3 ) )
            = ( ord_max_real @ X4 @ ( comple1385675409528146559p_real @ S3 ) ) ) ) ) ) ).

% Sup_insert_finite
thf(fact_7584_hom__Max__commute,axiom,
    ! [H2: code_integer > code_integer,N7: 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 @ N7 )
       => ( ( N7 != bot_bo3990330152332043303nteger )
         => ( ( H2 @ ( lattic4901227151466704046nteger @ N7 ) )
            = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7585_hom__Max__commute,axiom,
    ! [H2: real > real,N7: set_real] :
      ( ! [X3: real,Y3: real] :
          ( ( H2 @ ( ord_max_real @ X3 @ Y3 ) )
          = ( ord_max_real @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_real @ N7 )
       => ( ( N7 != bot_bot_set_real )
         => ( ( H2 @ ( lattic4275903605611617917x_real @ N7 ) )
            = ( lattic4275903605611617917x_real @ ( image_real_real @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7586_hom__Max__commute,axiom,
    ! [H2: $o > $o,N7: set_o] :
      ( ! [X3: $o,Y3: $o] :
          ( ( H2 @ ( ord_max_o @ X3 @ Y3 ) )
          = ( ord_max_o @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_o @ N7 )
       => ( ( N7 != bot_bot_set_o )
         => ( ( H2 @ ( lattic1921953407002678535_Max_o @ N7 ) )
            = ( lattic1921953407002678535_Max_o @ ( image_o_o @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7587_hom__Max__commute,axiom,
    ! [H2: extended_enat > extended_enat,N7: set_Extended_enat] :
      ( ! [X3: extended_enat,Y3: extended_enat] :
          ( ( H2 @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
          = ( ord_ma741700101516333627d_enat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite4001608067531595151d_enat @ N7 )
       => ( ( N7 != bot_bo7653980558646680370d_enat )
         => ( ( H2 @ ( lattic921264341876707157d_enat @ N7 ) )
            = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7588_hom__Max__commute,axiom,
    ! [H2: nat > nat,N7: set_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( H2 @ ( ord_max_nat @ X3 @ Y3 ) )
          = ( ord_max_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_nat @ N7 )
       => ( ( N7 != bot_bot_set_nat )
         => ( ( H2 @ ( lattic8265883725875713057ax_nat @ N7 ) )
            = ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7589_hom__Max__commute,axiom,
    ! [H2: int > int,N7: set_int] :
      ( ! [X3: int,Y3: int] :
          ( ( H2 @ ( ord_max_int @ X3 @ Y3 ) )
          = ( ord_max_int @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_int @ N7 )
       => ( ( N7 != bot_bot_set_int )
         => ( ( H2 @ ( lattic8263393255366662781ax_int @ N7 ) )
            = ( lattic8263393255366662781ax_int @ ( image_int_int @ H2 @ N7 ) ) ) ) ) ) ).

% hom_Max_commute
thf(fact_7590_Max_Osubset,axiom,
    ! [A3: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( B4 != bot_bo3990330152332043303nteger )
       => ( ( ord_le7084787975880047091nteger @ B4 @ A3 )
         => ( ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ B4 ) @ ( lattic4901227151466704046nteger @ A3 ) )
            = ( lattic4901227151466704046nteger @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7591_Max_Osubset,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( B4 != bot_bot_set_real )
       => ( ( ord_less_eq_set_real @ B4 @ A3 )
         => ( ( ord_max_real @ ( lattic4275903605611617917x_real @ B4 ) @ ( lattic4275903605611617917x_real @ A3 ) )
            = ( lattic4275903605611617917x_real @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7592_Max_Osubset,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( B4 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B4 @ A3 )
         => ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B4 ) @ ( lattic1921953407002678535_Max_o @ A3 ) )
            = ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7593_Max_Osubset,axiom,
    ! [A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( B4 != bot_bo7653980558646680370d_enat )
       => ( ( ord_le7203529160286727270d_enat @ B4 @ A3 )
         => ( ( ord_ma741700101516333627d_enat @ ( lattic921264341876707157d_enat @ B4 ) @ ( lattic921264341876707157d_enat @ A3 ) )
            = ( lattic921264341876707157d_enat @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7594_Max_Osubset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( B4 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B4 @ A3 )
         => ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B4 ) @ ( lattic8265883725875713057ax_nat @ A3 ) )
            = ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7595_Max_Osubset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( B4 != bot_bot_set_int )
       => ( ( ord_less_eq_set_int @ B4 @ A3 )
         => ( ( ord_max_int @ ( lattic8263393255366662781ax_int @ B4 ) @ ( lattic8263393255366662781ax_int @ A3 ) )
            = ( lattic8263393255366662781ax_int @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_7596_Max_Oclosed,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != 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 @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7597_Max_Oclosed,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != 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 @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7598_Max_Oclosed,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != 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 @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7599_Max_Oclosed,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X3: extended_enat,Y3: extended_enat] : ( member_Extended_enat @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) @ ( insert_Extended_enat @ X3 @ ( insert_Extended_enat @ Y3 @ bot_bo7653980558646680370d_enat ) ) )
         => ( member_Extended_enat @ ( lattic921264341876707157d_enat @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7600_Max_Oclosed,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != 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 @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7601_Max_Oclosed,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != 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 @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_7602_Max_Oinsert__not__elem,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ~ ( member_Code_integer @ X4 @ A3 )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X4 @ A3 ) )
            = ( ord_max_Code_integer @ X4 @ ( lattic4901227151466704046nteger @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7603_Max_Oinsert__not__elem,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ~ ( member_real @ X4 @ A3 )
       => ( ( A3 != bot_bot_set_real )
         => ( ( lattic4275903605611617917x_real @ ( insert_real @ X4 @ A3 ) )
            = ( ord_max_real @ X4 @ ( lattic4275903605611617917x_real @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7604_Max_Oinsert__not__elem,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X4 @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X4 @ A3 ) )
            = ( ord_max_o @ X4 @ ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7605_Max_Oinsert__not__elem,axiom,
    ! [A3: set_Extended_enat,X4: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ~ ( member_Extended_enat @ X4 @ A3 )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => ( ( lattic921264341876707157d_enat @ ( insert_Extended_enat @ X4 @ A3 ) )
            = ( ord_ma741700101516333627d_enat @ X4 @ ( lattic921264341876707157d_enat @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7606_Max_Oinsert__not__elem,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ X4 @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X4 @ A3 ) )
            = ( ord_max_nat @ X4 @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7607_Max_Oinsert__not__elem,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ~ ( member_int @ X4 @ A3 )
       => ( ( A3 != bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X4 @ A3 ) )
            = ( ord_max_int @ X4 @ ( lattic8263393255366662781ax_int @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_7608_Max_Ounion,axiom,
    ! [A3: set_Code_integer,B4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ( finite6017078050557962740nteger @ B4 )
         => ( ( B4 != bot_bo3990330152332043303nteger )
           => ( ( lattic4901227151466704046nteger @ ( sup_su848401254843788991nteger @ A3 @ B4 ) )
              = ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ A3 ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7609_Max_Ounion,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ( finite_finite_real @ B4 )
         => ( ( B4 != bot_bot_set_real )
           => ( ( lattic4275903605611617917x_real @ ( sup_sup_set_real @ A3 @ B4 ) )
              = ( ord_max_real @ ( lattic4275903605611617917x_real @ A3 ) @ ( lattic4275903605611617917x_real @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7610_Max_Ounion,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B4 )
         => ( ( B4 != bot_bot_set_o )
           => ( ( lattic1921953407002678535_Max_o @ ( sup_sup_set_o @ A3 @ B4 ) )
              = ( ord_max_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7611_Max_Ounion,axiom,
    ! [A3: set_Extended_enat,B4: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( finite4001608067531595151d_enat @ B4 )
         => ( ( B4 != bot_bo7653980558646680370d_enat )
           => ( ( lattic921264341876707157d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B4 ) )
              = ( ord_ma741700101516333627d_enat @ ( lattic921264341876707157d_enat @ A3 ) @ ( lattic921264341876707157d_enat @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7612_Max_Ounion,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ( B4 != bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A3 @ B4 ) )
              = ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7613_Max_Ounion,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B4 )
         => ( ( B4 != bot_bot_set_int )
           => ( ( lattic8263393255366662781ax_int @ ( sup_sup_set_int @ A3 @ B4 ) )
              = ( ord_max_int @ ( lattic8263393255366662781ax_int @ A3 ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_7614_conj__subset__def,axiom,
    ! [A3: set_complex,P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ A3
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le211207098394363844omplex @ A3 @ ( collect_complex @ P ) )
        & ( ord_le211207098394363844omplex @ A3 @ ( collect_complex @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7615_conj__subset__def,axiom,
    ! [A3: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A3
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ P ) )
        & ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7616_conj__subset__def,axiom,
    ! [A3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A3
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ P ) )
        & ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7617_conj__subset__def,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7618_conj__subset__def,axiom,
    ! [A3: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A3
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_int @ A3 @ ( collect_int @ P ) )
        & ( ord_less_eq_set_int @ A3 @ ( collect_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7619_Max_Oremove,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ X4 @ A3 )
       => ( ( ( ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) )
              = bot_bo3990330152332043303nteger )
           => ( ( lattic4901227151466704046nteger @ A3 )
              = X4 ) )
          & ( ( ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) )
             != bot_bo3990330152332043303nteger )
           => ( ( lattic4901227151466704046nteger @ A3 )
              = ( ord_max_Code_integer @ X4 @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7620_Max_Oremove,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X4 @ A3 )
       => ( ( ( ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) )
              = bot_bot_set_real )
           => ( ( lattic4275903605611617917x_real @ A3 )
              = X4 ) )
          & ( ( ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) )
             != bot_bot_set_real )
           => ( ( lattic4275903605611617917x_real @ A3 )
              = ( ord_max_real @ X4 @ ( lattic4275903605611617917x_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7621_Max_Oremove,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X4 @ A3 )
       => ( ( lattic1921953407002678535_Max_o @ A3 )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X4 )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( ord_max_o @ X4 @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7622_Max_Oremove,axiom,
    ! [A3: set_Extended_enat,X4: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X4 @ A3 )
       => ( ( ( ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) )
              = bot_bo7653980558646680370d_enat )
           => ( ( lattic921264341876707157d_enat @ A3 )
              = X4 ) )
          & ( ( ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) )
             != bot_bo7653980558646680370d_enat )
           => ( ( lattic921264341876707157d_enat @ A3 )
              = ( ord_ma741700101516333627d_enat @ X4 @ ( lattic921264341876707157d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7623_Max_Oremove,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X4 @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A3 )
              = X4 ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A3 )
              = ( ord_max_nat @ X4 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7624_Max_Oremove,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X4 @ A3 )
       => ( ( ( ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) )
              = bot_bot_set_int )
           => ( ( lattic8263393255366662781ax_int @ A3 )
              = X4 ) )
          & ( ( ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) )
             != bot_bot_set_int )
           => ( ( lattic8263393255366662781ax_int @ A3 )
              = ( ord_max_int @ X4 @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_7625_Max_Oinsert__remove,axiom,
    ! [A3: set_Code_integer,X4: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ( ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) )
            = bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X4 @ A3 ) )
            = X4 ) )
        & ( ( ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) )
           != bot_bo3990330152332043303nteger )
         => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X4 @ A3 ) )
            = ( ord_max_Code_integer @ X4 @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A3 @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7626_Max_Oinsert__remove,axiom,
    ! [A3: set_real,X4: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) )
            = bot_bot_set_real )
         => ( ( lattic4275903605611617917x_real @ ( insert_real @ X4 @ A3 ) )
            = X4 ) )
        & ( ( ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) )
           != bot_bot_set_real )
         => ( ( lattic4275903605611617917x_real @ ( insert_real @ X4 @ A3 ) )
            = ( ord_max_real @ X4 @ ( lattic4275903605611617917x_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7627_Max_Oinsert__remove,axiom,
    ! [A3: set_o,X4: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X4 @ A3 ) )
        = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X4 )
          & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( ord_max_o @ X4 @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7628_Max_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,X4: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) )
            = bot_bo7653980558646680370d_enat )
         => ( ( lattic921264341876707157d_enat @ ( insert_Extended_enat @ X4 @ A3 ) )
            = X4 ) )
        & ( ( ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) )
           != bot_bo7653980558646680370d_enat )
         => ( ( lattic921264341876707157d_enat @ ( insert_Extended_enat @ X4 @ A3 ) )
            = ( ord_ma741700101516333627d_enat @ X4 @ ( lattic921264341876707157d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7629_Max_Oinsert__remove,axiom,
    ! [A3: set_nat,X4: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X4 @ A3 ) )
            = X4 ) )
        & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X4 @ A3 ) )
            = ( ord_max_nat @ X4 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7630_Max_Oinsert__remove,axiom,
    ! [A3: set_int,X4: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) )
            = bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X4 @ A3 ) )
            = X4 ) )
        & ( ( ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) )
           != bot_bot_set_int )
         => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X4 @ A3 ) )
            = ( ord_max_int @ X4 @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_7631_Sup__option__def,axiom,
    ( comple8455683388168444585et_nat
    = ( ^ [A5: set_option_set_nat] :
          ( if_option_set_nat
          @ ( ( A5 = bot_bo5650944848895983264et_nat )
            | ( A5
              = ( insert6281397740311767046et_nat @ none_set_nat @ bot_bo5650944848895983264et_nat ) ) )
          @ none_set_nat
          @ ( some_set_nat @ ( comple7399068483239264473et_nat @ ( these_set_nat @ A5 ) ) ) ) ) ) ).

% Sup_option_def
thf(fact_7632_UN__constant__eq,axiom,
    ! [A: nat,A3: set_nat,F: nat > set_nat,C: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_7633_UN__constant__eq,axiom,
    ! [A: vEBT_VEBT,A3: set_VEBT_VEBT,F: vEBT_VEBT > set_nat,C: set_nat] :
      ( ( member_VEBT_VEBT @ A @ A3 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A3 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_6451421511446451829et_nat @ F @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_7634_UN__constant__eq,axiom,
    ! [A: real,A3: set_real,F: real > set_nat,C: set_nat] :
      ( ( member_real @ A @ A3 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A3 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_7635_UN__constant__eq,axiom,
    ! [A: int,A3: set_int,F: int > set_nat,C: set_nat] :
      ( ( member_int @ A @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_7636_UN__constant__eq,axiom,
    ! [A: set_nat,A3: set_set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( member_set_nat @ A @ A3 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_7637_these__not__empty__eq,axiom,
    ! [B4: set_op4508134149509766951at_nat] :
      ( ( ( these_8061402112538453438at_nat @ B4 )
       != bot_bo2099793752762293965at_nat )
      = ( ( B4 != bot_bo3414888551220300179at_nat )
        & ( B4
         != ( insert6842972033726296599at_nat @ none_P5556105721700978146at_nat @ bot_bo3414888551220300179at_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_7638_these__not__empty__eq,axiom,
    ! [B4: set_option_num] :
      ( ( ( these_num @ B4 )
       != bot_bot_set_num )
      = ( ( B4 != bot_bo725063100934353204on_num )
        & ( B4
         != ( insert_option_num @ none_num @ bot_bo725063100934353204on_num ) ) ) ) ).

% these_not_empty_eq
thf(fact_7639_these__not__empty__eq,axiom,
    ! [B4: set_option_real] :
      ( ( ( these_real @ B4 )
       != bot_bot_set_real )
      = ( ( B4 != bot_bo677810143523618246n_real )
        & ( B4
         != ( insert_option_real @ none_real @ bot_bo677810143523618246n_real ) ) ) ) ).

% these_not_empty_eq
thf(fact_7640_these__not__empty__eq,axiom,
    ! [B4: set_option_o] :
      ( ( ( these_o @ B4 )
       != bot_bot_set_o )
      = ( ( B4 != bot_bot_set_option_o )
        & ( B4
         != ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_not_empty_eq
thf(fact_7641_these__not__empty__eq,axiom,
    ! [B4: set_option_nat] :
      ( ( ( these_nat @ B4 )
       != bot_bot_set_nat )
      = ( ( B4 != bot_bo5009843511495006442on_nat )
        & ( B4
         != ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_7642_these__not__empty__eq,axiom,
    ! [B4: set_option_int] :
      ( ( ( these_int @ B4 )
       != bot_bot_set_int )
      = ( ( B4 != bot_bo2519760366563279686on_int )
        & ( B4
         != ( insert_option_int @ none_int @ bot_bo2519760366563279686on_int ) ) ) ) ).

% these_not_empty_eq
thf(fact_7643_these__empty__eq,axiom,
    ! [B4: set_op4508134149509766951at_nat] :
      ( ( ( these_8061402112538453438at_nat @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ( B4 = bot_bo3414888551220300179at_nat )
        | ( B4
          = ( insert6842972033726296599at_nat @ none_P5556105721700978146at_nat @ bot_bo3414888551220300179at_nat ) ) ) ) ).

% these_empty_eq
thf(fact_7644_these__empty__eq,axiom,
    ! [B4: set_option_num] :
      ( ( ( these_num @ B4 )
        = bot_bot_set_num )
      = ( ( B4 = bot_bo725063100934353204on_num )
        | ( B4
          = ( insert_option_num @ none_num @ bot_bo725063100934353204on_num ) ) ) ) ).

% these_empty_eq
thf(fact_7645_these__empty__eq,axiom,
    ! [B4: set_option_real] :
      ( ( ( these_real @ B4 )
        = bot_bot_set_real )
      = ( ( B4 = bot_bo677810143523618246n_real )
        | ( B4
          = ( insert_option_real @ none_real @ bot_bo677810143523618246n_real ) ) ) ) ).

% these_empty_eq
thf(fact_7646_these__empty__eq,axiom,
    ! [B4: set_option_o] :
      ( ( ( these_o @ B4 )
        = bot_bot_set_o )
      = ( ( B4 = bot_bot_set_option_o )
        | ( B4
          = ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_empty_eq
thf(fact_7647_these__empty__eq,axiom,
    ! [B4: set_option_nat] :
      ( ( ( these_nat @ B4 )
        = bot_bot_set_nat )
      = ( ( B4 = bot_bo5009843511495006442on_nat )
        | ( B4
          = ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).

% these_empty_eq
thf(fact_7648_these__empty__eq,axiom,
    ! [B4: set_option_int] :
      ( ( ( these_int @ B4 )
        = bot_bot_set_int )
      = ( ( B4 = bot_bo2519760366563279686on_int )
        | ( B4
          = ( insert_option_int @ none_int @ bot_bo2519760366563279686on_int ) ) ) ) ).

% these_empty_eq
thf(fact_7649_these__insert__Some,axiom,
    ! [X4: vEBT_VEBT,A3: set_option_VEBT_VEBT] :
      ( ( these_VEBT_VEBT @ ( insert9052096497370831178T_VEBT @ ( some_VEBT_VEBT @ X4 ) @ A3 ) )
      = ( insert_VEBT_VEBT @ X4 @ ( these_VEBT_VEBT @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7650_these__insert__Some,axiom,
    ! [X4: int,A3: set_option_int] :
      ( ( these_int @ ( insert_option_int @ ( some_int @ X4 ) @ A3 ) )
      = ( insert_int @ X4 @ ( these_int @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7651_these__insert__Some,axiom,
    ! [X4: $o,A3: set_option_o] :
      ( ( these_o @ ( insert_option_o @ ( some_o @ X4 ) @ A3 ) )
      = ( insert_o @ X4 @ ( these_o @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7652_these__insert__Some,axiom,
    ! [X4: real,A3: set_option_real] :
      ( ( these_real @ ( insert_option_real @ ( some_real @ X4 ) @ A3 ) )
      = ( insert_real @ X4 @ ( these_real @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7653_these__insert__Some,axiom,
    ! [X4: product_prod_nat_nat,A3: set_op4508134149509766951at_nat] :
      ( ( these_8061402112538453438at_nat @ ( insert6842972033726296599at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ A3 ) )
      = ( insert8211810215607154385at_nat @ X4 @ ( these_8061402112538453438at_nat @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7654_these__insert__Some,axiom,
    ! [X4: nat,A3: set_option_nat] :
      ( ( these_nat @ ( insert_option_nat @ ( some_nat @ X4 ) @ A3 ) )
      = ( insert_nat @ X4 @ ( these_nat @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7655_these__insert__Some,axiom,
    ! [X4: num,A3: set_option_num] :
      ( ( these_num @ ( insert_option_num @ ( some_num @ X4 ) @ A3 ) )
      = ( insert_num @ X4 @ ( these_num @ A3 ) ) ) ).

% these_insert_Some
thf(fact_7656_these__image__Some__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( these_8061402112538453438at_nat @ ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_7657_these__image__Some__eq,axiom,
    ! [A3: set_nat] :
      ( ( these_nat @ ( image_nat_option_nat @ some_nat @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_7658_these__image__Some__eq,axiom,
    ! [A3: set_num] :
      ( ( these_num @ ( image_num_option_num @ some_num @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_7659_these__empty,axiom,
    ( ( these_real @ bot_bo677810143523618246n_real )
    = bot_bot_set_real ) ).

% these_empty
thf(fact_7660_these__empty,axiom,
    ( ( these_o @ bot_bot_set_option_o )
    = bot_bot_set_o ) ).

% these_empty
thf(fact_7661_these__empty,axiom,
    ( ( these_nat @ bot_bo5009843511495006442on_nat )
    = bot_bot_set_nat ) ).

% these_empty
thf(fact_7662_these__empty,axiom,
    ( ( these_int @ bot_bo2519760366563279686on_int )
    = bot_bot_set_int ) ).

% these_empty
thf(fact_7663_these__insert__None,axiom,
    ! [A3: set_option_nat] :
      ( ( these_nat @ ( insert_option_nat @ none_nat @ A3 ) )
      = ( these_nat @ A3 ) ) ).

% these_insert_None
thf(fact_7664_these__insert__None,axiom,
    ! [A3: set_op4508134149509766951at_nat] :
      ( ( these_8061402112538453438at_nat @ ( insert6842972033726296599at_nat @ none_P5556105721700978146at_nat @ A3 ) )
      = ( these_8061402112538453438at_nat @ A3 ) ) ).

% these_insert_None
thf(fact_7665_these__insert__None,axiom,
    ! [A3: set_option_num] :
      ( ( these_num @ ( insert_option_num @ none_num @ A3 ) )
      = ( these_num @ A3 ) ) ).

% these_insert_None
thf(fact_7666_in__these__eq,axiom,
    ! [X4: vEBT_VEBT,A3: set_option_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( these_VEBT_VEBT @ A3 ) )
      = ( member2458453091852628771T_VEBT @ ( some_VEBT_VEBT @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7667_in__these__eq,axiom,
    ! [X4: real,A3: set_option_real] :
      ( ( member_real @ X4 @ ( these_real @ A3 ) )
      = ( member_option_real @ ( some_real @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7668_in__these__eq,axiom,
    ! [X4: int,A3: set_option_int] :
      ( ( member_int @ X4 @ ( these_int @ A3 ) )
      = ( member_option_int @ ( some_int @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7669_in__these__eq,axiom,
    ! [X4: set_nat,A3: set_option_set_nat] :
      ( ( member_set_nat @ X4 @ ( these_set_nat @ A3 ) )
      = ( member8989860449721436141et_nat @ ( some_set_nat @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7670_in__these__eq,axiom,
    ! [X4: product_prod_nat_nat,A3: set_op4508134149509766951at_nat] :
      ( ( member8440522571783428010at_nat @ X4 @ ( these_8061402112538453438at_nat @ A3 ) )
      = ( member3954567711264315760at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7671_in__these__eq,axiom,
    ! [X4: nat,A3: set_option_nat] :
      ( ( member_nat @ X4 @ ( these_nat @ A3 ) )
      = ( member_option_nat @ ( some_nat @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7672_in__these__eq,axiom,
    ! [X4: num,A3: set_option_num] :
      ( ( member_num @ X4 @ ( these_num @ A3 ) )
      = ( member_option_num @ ( some_num @ X4 ) @ A3 ) ) ).

% in_these_eq
thf(fact_7673_Some__image__these__eq,axiom,
    ! [A3: set_op4508134149509766951at_nat] :
      ( ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ ( these_8061402112538453438at_nat @ A3 ) )
      = ( collec5929226652830133682at_nat
        @ ^ [X: option4927543243414619207at_nat] :
            ( ( member3954567711264315760at_nat @ X @ A3 )
            & ( X != none_P5556105721700978146at_nat ) ) ) ) ).

% Some_image_these_eq
thf(fact_7674_Some__image__these__eq,axiom,
    ! [A3: set_option_nat] :
      ( ( image_nat_option_nat @ some_nat @ ( these_nat @ A3 ) )
      = ( collect_option_nat
        @ ^ [X: option_nat] :
            ( ( member_option_nat @ X @ A3 )
            & ( X != none_nat ) ) ) ) ).

% Some_image_these_eq
thf(fact_7675_Some__image__these__eq,axiom,
    ! [A3: set_option_num] :
      ( ( image_num_option_num @ some_num @ ( these_num @ A3 ) )
      = ( collect_option_num
        @ ^ [X: option_num] :
            ( ( member_option_num @ X @ A3 )
            & ( X != none_num ) ) ) ) ).

% Some_image_these_eq
thf(fact_7676_dual__Min,axiom,
    ( ( lattices_Min_rat
      @ ^ [X: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ X ) )
    = lattic7630753665789217321ax_rat ) ).

% dual_Min
thf(fact_7677_dual__Min,axiom,
    ( ( lattices_Min_num
      @ ^ [X: num,Y4: num] : ( ord_less_eq_num @ Y4 @ X ) )
    = lattic4823215512031491691ax_num ) ).

% dual_Min
thf(fact_7678_dual__Min,axiom,
    ( ( lattices_Min_nat
      @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X ) )
    = lattic8265883725875713057ax_nat ) ).

% dual_Min
thf(fact_7679_dual__Min,axiom,
    ( ( lattices_Min_int
      @ ^ [X: int,Y4: int] : ( ord_less_eq_int @ Y4 @ X ) )
    = lattic8263393255366662781ax_int ) ).

% dual_Min
thf(fact_7680_Un__set__drop__extend,axiom,
    ! [J: nat,L: list_set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ J )
     => ( ( ord_less_nat @ J @ ( size_s3254054031482475050et_nat @ L ) )
       => ( ( sup_sup_set_nat @ ( nth_set_nat @ L @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) @ ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( drop_set_nat @ J @ L ) ) ) )
          = ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( drop_set_nat @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) @ L ) ) ) ) ) ) ).

% Un_set_drop_extend
thf(fact_7681_subset__subseqs,axiom,
    ! [X7: set_VEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X7 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( member_set_VEBT_VEBT @ X7 @ ( image_6463372868993444447T_VEBT @ set_VEBT_VEBT2 @ ( set_list_VEBT_VEBT2 @ ( subseqs_VEBT_VEBT @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7682_subset__subseqs,axiom,
    ! [X7: set_nat,Xs2: list_nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( set_nat2 @ Xs2 ) )
     => ( member_set_nat @ X7 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7683_subset__subseqs,axiom,
    ! [X7: set_real,Xs2: list_real] :
      ( ( ord_less_eq_set_real @ X7 @ ( set_real2 @ Xs2 ) )
     => ( member_set_real @ X7 @ ( image_6239767680843085477t_real @ set_real2 @ ( set_list_real2 @ ( subseqs_real @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7684_subset__subseqs,axiom,
    ! [X7: set_o,Xs2: list_o] :
      ( ( ord_less_eq_set_o @ X7 @ ( set_o2 @ Xs2 ) )
     => ( member_set_o @ X7 @ ( image_list_o_set_o @ set_o2 @ ( set_list_o2 @ ( subseqs_o @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7685_subset__subseqs,axiom,
    ! [X7: set_int,Xs2: list_int] :
      ( ( ord_less_eq_set_int @ X7 @ ( set_int2 @ Xs2 ) )
     => ( member_set_int @ X7 @ ( image_3606813740839090725et_int @ set_int2 @ ( set_list_int2 @ ( subseqs_int @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7686_image__Fpow__mono,axiom,
    ! [F: nat > real,A3: set_nat,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ A3 ) @ B4 )
     => ( ord_le3558479182127378552t_real @ ( image_6333053925516494319t_real @ ( image_nat_real @ F ) @ ( finite_Fpow_nat @ A3 ) ) @ ( finite_Fpow_real @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_7687_image__Fpow__mono,axiom,
    ! [F: nat > set_nat,A3: set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ B4 )
     => ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A3 ) ) @ ( finite_Fpow_set_nat @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_7688_image__Fpow__mono,axiom,
    ! [F: nat > nat,A3: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B4 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A3 ) ) @ ( finite_Fpow_nat @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_7689_image__Fpow__mono,axiom,
    ! [F: nat > int,A3: set_nat,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B4 )
     => ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A3 ) ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_7690_image__Fpow__mono,axiom,
    ! [F: int > int,A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ B4 )
     => ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A3 ) ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_7691_set__list__bind,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( bind_V4754371891896957063T_VEBT @ Xs2 @ F ) )
      = ( comple2820511241208326657T_VEBT
        @ ( image_2685870239581809509T_VEBT
          @ ^ [X: vEBT_VEBT] : ( set_VEBT_VEBT2 @ ( F @ X ) )
          @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7692_set__list__bind,axiom,
    ! [Xs2: list_nat,F: nat > list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( bind_nat_VEBT_VEBT @ Xs2 @ F ) )
      = ( comple2820511241208326657T_VEBT
        @ ( image_1406951880692228733T_VEBT
          @ ^ [X: nat] : ( set_VEBT_VEBT2 @ ( F @ X ) )
          @ ( set_nat2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7693_set__list__bind,axiom,
    ! [Xs2: list_real,F: real > list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( bind_real_VEBT_VEBT @ Xs2 @ F ) )
      = ( comple2820511241208326657T_VEBT
        @ ( image_6925917818215209377T_VEBT
          @ ^ [X: real] : ( set_VEBT_VEBT2 @ ( F @ X ) )
          @ ( set_real2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7694_set__list__bind,axiom,
    ! [Xs2: list_o,F: $o > list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( bind_o_VEBT_VEBT @ Xs2 @ F ) )
      = ( comple2820511241208326657T_VEBT
        @ ( image_7704241249472752129T_VEBT
          @ ^ [X: $o] : ( set_VEBT_VEBT2 @ ( F @ X ) )
          @ ( set_o2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7695_set__list__bind,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > list_real] :
      ( ( set_real2 @ ( bind_VEBT_VEBT_real @ Xs2 @ F ) )
      = ( comple3096694443085538997t_real
        @ ( image_6636839513470643793t_real
          @ ^ [X: vEBT_VEBT] : ( set_real2 @ ( F @ X ) )
          @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7696_set__list__bind,axiom,
    ! [Xs2: list_nat,F: nat > list_real] :
      ( ( set_real2 @ ( bind_nat_real @ Xs2 @ F ) )
      = ( comple3096694443085538997t_real
        @ ( image_nat_set_real
          @ ^ [X: nat] : ( set_real2 @ ( F @ X ) )
          @ ( set_nat2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7697_set__list__bind,axiom,
    ! [Xs2: list_real,F: real > list_real] :
      ( ( set_real2 @ ( bind_real_real @ Xs2 @ F ) )
      = ( comple3096694443085538997t_real
        @ ( image_real_set_real
          @ ^ [X: real] : ( set_real2 @ ( F @ X ) )
          @ ( set_real2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7698_set__list__bind,axiom,
    ! [Xs2: list_o,F: $o > list_real] :
      ( ( set_real2 @ ( bind_o_real @ Xs2 @ F ) )
      = ( comple3096694443085538997t_real
        @ ( image_o_set_real
          @ ^ [X: $o] : ( set_real2 @ ( F @ X ) )
          @ ( set_o2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7699_set__list__bind,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > list_o] :
      ( ( set_o2 @ ( bind_VEBT_VEBT_o @ Xs2 @ F ) )
      = ( comple90263536869209701_set_o
        @ ( image_7883550159813902793_set_o
          @ ^ [X: vEBT_VEBT] : ( set_o2 @ ( F @ X ) )
          @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7700_set__list__bind,axiom,
    ! [Xs2: list_nat,F: nat > list_o] :
      ( ( set_o2 @ ( bind_nat_o @ Xs2 @ F ) )
      = ( comple90263536869209701_set_o
        @ ( image_nat_set_o
          @ ^ [X: nat] : ( set_o2 @ ( F @ X ) )
          @ ( set_nat2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_7701_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7702_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7703_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7704_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7705_power__shift,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X4 @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X4 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_7706_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_7707_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_7708_nat__power__eq__Suc__0__iff,axiom,
    ! [X4: nat,M: nat] :
      ( ( ( power_power_nat @ X4 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X4
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_7709_nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_7710_drop0,axiom,
    ( ( drop_o @ zero_zero_nat )
    = ( ^ [X: list_o] : X ) ) ).

% drop0
thf(fact_7711_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_7712_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_7713_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_7714_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_7715_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_7716_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_7717_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_7718_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_7719_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_7720_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_7721_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_7722_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_7723_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_7724_drop__Suc__Cons,axiom,
    ! [N: nat,X4: $o,Xs2: list_o] :
      ( ( drop_o @ ( suc @ N ) @ ( cons_o @ X4 @ Xs2 ) )
      = ( drop_o @ N @ Xs2 ) ) ).

% drop_Suc_Cons
thf(fact_7725_drop__Suc__Cons,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X4 @ Xs2 ) )
      = ( drop_nat @ N @ Xs2 ) ) ).

% drop_Suc_Cons
thf(fact_7726_drop__Suc__Cons,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( drop_int @ ( suc @ N ) @ ( cons_int @ X4 @ Xs2 ) )
      = ( drop_int @ N @ Xs2 ) ) ).

% drop_Suc_Cons
thf(fact_7727_length__drop,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( size_size_list_real @ ( drop_real @ N @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ).

% length_drop
thf(fact_7728_length__drop,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( size_size_list_o @ ( drop_o @ N @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ).

% length_drop
thf(fact_7729_length__drop,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% length_drop
thf(fact_7730_length__drop,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( size_size_list_int @ ( drop_int @ N @ Xs2 ) )
      = ( minus_minus_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).

% length_drop
thf(fact_7731_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_o,X4: $o] :
      ( ( ord_less_nat @ M @ N )
     => ( ( drop_o @ N @ ( list_update_o @ L @ M @ X4 ) )
        = ( drop_o @ N @ L ) ) ) ).

% drop_upd_irrelevant
thf(fact_7732_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ M @ N )
     => ( ( drop_VEBT_VEBTi @ N @ ( list_u6098035379799741383_VEBTi @ L @ M @ X4 ) )
        = ( drop_VEBT_VEBTi @ N @ L ) ) ) ).

% drop_upd_irrelevant
thf(fact_7733_drop__upd__irrelevant,axiom,
    ! [M: nat,N: nat,L: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ M @ N )
     => ( ( drop_VEBT_VEBT @ N @ ( list_u1324408373059187874T_VEBT @ L @ M @ X4 ) )
        = ( drop_VEBT_VEBT @ N @ L ) ) ) ).

% drop_upd_irrelevant
thf(fact_7734_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_o,X4: $o] :
      ( ( ord_less_nat @ N @ M )
     => ( ( drop_o @ M @ ( list_update_o @ Xs2 @ N @ X4 ) )
        = ( drop_o @ M @ Xs2 ) ) ) ).

% drop_update_cancel
thf(fact_7735_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ M )
     => ( ( drop_VEBT_VEBTi @ M @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N @ X4 ) )
        = ( drop_VEBT_VEBTi @ M @ Xs2 ) ) ) ).

% drop_update_cancel
thf(fact_7736_drop__update__cancel,axiom,
    ! [N: nat,M: nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ M )
     => ( ( drop_VEBT_VEBT @ M @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X4 ) )
        = ( drop_VEBT_VEBT @ M @ Xs2 ) ) ) ).

% drop_update_cancel
thf(fact_7737_power__strict__increasing__iff,axiom,
    ! [B: real,X4: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X4 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_7738_power__strict__increasing__iff,axiom,
    ! [B: rat,X4: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X4 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_7739_power__strict__increasing__iff,axiom,
    ! [B: nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_7740_power__strict__increasing__iff,axiom,
    ! [B: int,X4: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_7741_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7742_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7743_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7744_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7745_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_7746_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_7747_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_7748_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_7749_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_7750_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_7751_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_7752_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_7753_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_7754_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7755_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7756_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7757_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7758_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7759_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7760_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7761_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7762_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7763_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7764_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7765_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X4: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7766_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7767_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7768_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7769_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7770_last__drop,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( ( last_real @ ( drop_real @ N @ Xs2 ) )
        = ( last_real @ Xs2 ) ) ) ).

% last_drop
thf(fact_7771_last__drop,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ( last_o @ ( drop_o @ N @ Xs2 ) )
        = ( last_o @ Xs2 ) ) ) ).

% last_drop
thf(fact_7772_last__drop,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( last_nat @ ( drop_nat @ N @ Xs2 ) )
        = ( last_nat @ Xs2 ) ) ) ).

% last_drop
thf(fact_7773_last__drop,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ( last_int @ ( drop_int @ N @ Xs2 ) )
        = ( last_int @ Xs2 ) ) ) ).

% last_drop
thf(fact_7774_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_7775_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_7776_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_7777_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_7778_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_7779_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_7780_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_7781_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_7782_power__increasing__iff,axiom,
    ! [B: real,X4: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X4 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_7783_power__increasing__iff,axiom,
    ! [B: rat,X4: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X4 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_7784_power__increasing__iff,axiom,
    ! [B: nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_7785_power__increasing__iff,axiom,
    ! [B: int,X4: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_7786_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7787_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7788_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7789_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7790_nth__drop,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBTi,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( drop_VEBT_VEBTi @ N @ Xs2 ) @ I )
        = ( nth_VEBT_VEBTi @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7791_nth__drop,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( drop_VEBT_VEBT @ N @ Xs2 ) @ I )
        = ( nth_VEBT_VEBT @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7792_nth__drop,axiom,
    ! [N: nat,Xs2: list_real,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_real @ ( drop_real @ N @ Xs2 ) @ I )
        = ( nth_real @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7793_nth__drop,axiom,
    ! [N: nat,Xs2: list_o,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( drop_o @ N @ Xs2 ) @ I )
        = ( nth_o @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7794_nth__drop,axiom,
    ! [N: nat,Xs2: list_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( drop_nat @ N @ Xs2 ) @ I )
        = ( nth_nat @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7795_nth__drop,axiom,
    ! [N: nat,Xs2: list_int,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( drop_int @ N @ Xs2 ) @ I )
        = ( nth_int @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_7796_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_7797_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_7798_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_7799_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_7800_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7801_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7802_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7803_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7804_drop__0,axiom,
    ! [Xs2: list_o] :
      ( ( drop_o @ zero_zero_nat @ Xs2 )
      = Xs2 ) ).

% drop_0
thf(fact_7805_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_7806_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_7807_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_7808_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_7809_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_7810_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_7811_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_7812_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_7813_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_7814_power__commutes,axiom,
    ! [A: assn,N: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A @ N ) @ A )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).

% power_commutes
thf(fact_7815_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_7816_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7817_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7818_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7819_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7820_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A @ B ) @ N )
      = ( times_times_assn @ ( power_power_assn @ A @ N ) @ ( power_power_assn @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7821_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_7822_power__commuting__commutes,axiom,
    ! [X4: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X4 @ Y )
        = ( times_times_real @ Y @ X4 ) )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7823_power__commuting__commutes,axiom,
    ! [X4: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X4 @ Y )
        = ( times_times_rat @ Y @ X4 ) )
     => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7824_power__commuting__commutes,axiom,
    ! [X4: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X4 @ Y )
        = ( times_times_nat @ Y @ X4 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7825_power__commuting__commutes,axiom,
    ! [X4: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X4 @ Y )
        = ( times_times_int @ Y @ X4 ) )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7826_power__commuting__commutes,axiom,
    ! [X4: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X4 @ Y )
        = ( times_times_assn @ Y @ X4 ) )
     => ( ( times_times_assn @ ( power_power_assn @ X4 @ N ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7827_power__commuting__commutes,axiom,
    ! [X4: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X4 @ Y )
        = ( times_times_complex @ Y @ X4 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_7828_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_7829_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_7830_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_7831_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_7832_drop__eq__ConsD,axiom,
    ! [N: nat,Xs2: list_o,X4: $o,Xs4: list_o] :
      ( ( ( drop_o @ N @ Xs2 )
        = ( cons_o @ X4 @ Xs4 ) )
     => ( ( drop_o @ ( suc @ N ) @ Xs2 )
        = Xs4 ) ) ).

% drop_eq_ConsD
thf(fact_7833_drop__eq__ConsD,axiom,
    ! [N: nat,Xs2: list_nat,X4: nat,Xs4: list_nat] :
      ( ( ( drop_nat @ N @ Xs2 )
        = ( cons_nat @ X4 @ Xs4 ) )
     => ( ( drop_nat @ ( suc @ N ) @ Xs2 )
        = Xs4 ) ) ).

% drop_eq_ConsD
thf(fact_7834_drop__eq__ConsD,axiom,
    ! [N: nat,Xs2: list_int,X4: int,Xs4: list_int] :
      ( ( ( drop_int @ N @ Xs2 )
        = ( cons_int @ X4 @ Xs4 ) )
     => ( ( drop_int @ ( suc @ N ) @ Xs2 )
        = Xs4 ) ) ).

% drop_eq_ConsD
thf(fact_7835_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( drop_VEBT_VEBT @ N @ Xs2 ) ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_7836_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_7837_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ ( drop_real @ N @ Xs2 ) ) @ ( set_real2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_7838_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( drop_o @ N @ Xs2 ) ) @ ( set_o2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_7839_set__drop__subset,axiom,
    ! [N: nat,Xs2: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( drop_int @ N @ Xs2 ) ) @ ( set_int2 @ Xs2 ) ) ).

% set_drop_subset
thf(fact_7840_sorted__drop,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( sorted_wrt_o @ ord_less_eq_o @ ( drop_o @ N @ Xs2 ) ) ) ).

% sorted_drop
thf(fact_7841_sorted__drop,axiom,
    ! [Xs2: list_rat,N: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ ( drop_rat @ N @ Xs2 ) ) ) ).

% sorted_drop
thf(fact_7842_sorted__drop,axiom,
    ! [Xs2: list_num,N: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( sorted_wrt_num @ ord_less_eq_num @ ( drop_num @ N @ Xs2 ) ) ) ).

% sorted_drop
thf(fact_7843_sorted__drop,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs2 ) ) ) ).

% sorted_drop
thf(fact_7844_sorted__drop,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( drop_int @ N @ Xs2 ) ) ) ).

% sorted_drop
thf(fact_7845_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_7846_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_7847_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_7848_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_7849_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_7850_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_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_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_7859_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_7860_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_7861_empty__in__Fpow,axiom,
    ! [A3: set_real] : ( member_set_real @ bot_bot_set_real @ ( finite_Fpow_real @ A3 ) ) ).

% empty_in_Fpow
thf(fact_7862_empty__in__Fpow,axiom,
    ! [A3: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A3 ) ) ).

% empty_in_Fpow
thf(fact_7863_empty__in__Fpow,axiom,
    ! [A3: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A3 ) ) ).

% empty_in_Fpow
thf(fact_7864_empty__in__Fpow,axiom,
    ! [A3: set_int] : ( member_set_int @ bot_bot_set_int @ ( finite_Fpow_int @ A3 ) ) ).

% empty_in_Fpow
thf(fact_7865_left__right__inverse__power,axiom,
    ! [X4: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X4 @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_7866_left__right__inverse__power,axiom,
    ! [X4: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X4 @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X4 @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_7867_left__right__inverse__power,axiom,
    ! [X4: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X4 @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_7868_left__right__inverse__power,axiom,
    ! [X4: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X4 @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_7869_left__right__inverse__power,axiom,
    ! [X4: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X4 @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X4 @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_7870_left__right__inverse__power,axiom,
    ! [X4: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X4 @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_7871_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_7872_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_7873_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_7874_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_7875_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_7876_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_7877_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_7878_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_7879_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_7880_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_7881_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_7882_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_7883_power__0,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_7884_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_7885_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_7886_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_7887_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_7888_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_7889_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_7890_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_7891_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_7892_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_7893_power__add,axiom,
    ! [A: assn,M: nat,N: nat] :
      ( ( power_power_assn @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A @ M ) @ ( power_power_assn @ A @ N ) ) ) ).

% power_add
thf(fact_7894_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_7895_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_7896_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( drop_VEBT_VEBT @ M @ Xs2 ) ) @ ( set_VEBT_VEBT2 @ ( drop_VEBT_VEBT @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_7897_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_7898_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_real] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_set_real @ ( set_real2 @ ( drop_real @ M @ Xs2 ) ) @ ( set_real2 @ ( drop_real @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_7899_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_set_o @ ( set_o2 @ ( drop_o @ M @ Xs2 ) ) @ ( set_o2 @ ( drop_o @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_7900_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_less_eq_set_int @ ( set_int2 @ ( drop_int @ M @ Xs2 ) ) @ ( set_int2 @ ( drop_int @ N @ Xs2 ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_7901_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs2: list_o,X4: $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( drop_o @ M @ ( list_update_o @ Xs2 @ N @ X4 ) )
        = ( list_update_o @ ( drop_o @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X4 ) ) ) ).

% drop_update_swap
thf(fact_7902_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( drop_VEBT_VEBTi @ M @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N @ X4 ) )
        = ( list_u6098035379799741383_VEBTi @ ( drop_VEBT_VEBTi @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X4 ) ) ) ).

% drop_update_swap
thf(fact_7903_drop__update__swap,axiom,
    ! [M: nat,N: nat,Xs2: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( drop_VEBT_VEBT @ M @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X4 ) )
        = ( list_u1324408373059187874T_VEBT @ ( drop_VEBT_VEBT @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X4 ) ) ) ).

% drop_update_swap
thf(fact_7904_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_7905_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_7906_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_7907_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_7908_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_7909_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_7910_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_7911_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_7912_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_7913_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_7914_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_7915_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_7916_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_7917_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_7918_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_7919_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_7920_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_7921_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_7922_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_7923_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_7924_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_7925_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_7926_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_7927_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_7928_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_7929_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_7930_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_7931_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_7932_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_7933_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_7934_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_7935_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_7936_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_7937_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_7938_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_7939_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_7940_power__strict__increasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_7941_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_7942_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_7943_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_7944_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_7945_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_7946_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_7947_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_7948_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_7949_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_7950_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_7951_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_7952_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_7953_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_7954_power__increasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_7955_power__increasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_7956_power__increasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_7957_power__increasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_7958_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7959_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7960_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7961_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7962_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_7963_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_7964_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_7965_real__arch__pow__inv,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_7966_drop__Cons_H,axiom,
    ! [N: nat,X4: $o,Xs2: list_o] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_o @ N @ ( cons_o @ X4 @ Xs2 ) )
          = ( cons_o @ X4 @ Xs2 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_o @ N @ ( cons_o @ X4 @ Xs2 ) )
          = ( drop_o @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).

% drop_Cons'
thf(fact_7967_drop__Cons_H,axiom,
    ! [N: nat,X4: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X4 @ Xs2 ) )
          = ( cons_nat @ X4 @ Xs2 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X4 @ Xs2 ) )
          = ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).

% drop_Cons'
thf(fact_7968_drop__Cons_H,axiom,
    ! [N: nat,X4: int,Xs2: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_int @ N @ ( cons_int @ X4 @ Xs2 ) )
          = ( cons_int @ X4 @ Xs2 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_int @ N @ ( cons_int @ X4 @ Xs2 ) )
          = ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).

% drop_Cons'
thf(fact_7969_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_7970_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_7971_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_7972_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_7973_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_7974_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_7975_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_7976_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_7977_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_7978_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_7979_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_7980_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_7981_length__n__lists,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( size_s6660260683639930848t_real @ ( n_lists_real @ N @ Xs2 ) )
      = ( power_power_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ).

% length_n_lists
thf(fact_7982_length__n__lists,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( size_s2710708370519433104list_o @ ( n_lists_o @ N @ Xs2 ) )
      = ( power_power_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ).

% length_n_lists
thf(fact_7983_length__n__lists,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N @ Xs2 ) )
      = ( power_power_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% length_n_lists
thf(fact_7984_length__n__lists,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( size_s533118279054570080st_int @ ( n_lists_int @ N @ Xs2 ) )
      = ( power_power_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).

% length_n_lists
thf(fact_7985_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_7986_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_7987_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_7988_power__strict__decreasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_7989_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_7990_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_7991_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_7992_power__decreasing,axiom,
    ! [N: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_7993_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_7994_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_7995_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_7996_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_7997_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_7998_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_7999_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_8000_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_8001_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_8002_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_8003_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_8004_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_8005_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_8006_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_8007_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_8008_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_8009_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_8010_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_8011_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_8012_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_8013_Fpow__mono,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B4 )
     => ( ord_le4403425263959731960et_int @ ( finite_Fpow_int @ A3 ) @ ( finite_Fpow_int @ B4 ) ) ) ).

% Fpow_mono
thf(fact_8014_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_8015_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_8016_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_8017_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A5: set_nat] :
          ( collect_set_nat
          @ ^ [X8: set_nat] :
              ( ( ord_less_eq_set_nat @ X8 @ A5 )
              & ( finite_finite_nat @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_8018_Fpow__def,axiom,
    ( finite_Fpow_complex
    = ( ^ [A5: set_complex] :
          ( collect_set_complex
          @ ^ [X8: set_complex] :
              ( ( ord_le211207098394363844omplex @ X8 @ A5 )
              & ( finite3207457112153483333omplex @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_8019_Fpow__def,axiom,
    ( finite1532502677820914807nteger
    = ( ^ [A5: set_Code_integer] :
          ( collec574505750873337192nteger
          @ ^ [X8: set_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ X8 @ A5 )
              & ( finite6017078050557962740nteger @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_8020_Fpow__def,axiom,
    ( finite_Fpow_int
    = ( ^ [A5: set_int] :
          ( collect_set_int
          @ ^ [X8: set_int] :
              ( ( ord_less_eq_set_int @ X8 @ A5 )
              & ( finite_finite_int @ X8 ) ) ) ) ) ).

% Fpow_def
thf(fact_8021_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( cons_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ I ) @ ( drop_VEBT_VEBTi @ ( suc @ I ) @ Xs2 ) )
        = ( drop_VEBT_VEBTi @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8022_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( cons_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( drop_VEBT_VEBT @ ( suc @ I ) @ Xs2 ) )
        = ( drop_VEBT_VEBT @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8023_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( cons_real @ ( nth_real @ Xs2 @ I ) @ ( drop_real @ ( suc @ I ) @ Xs2 ) )
        = ( drop_real @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8024_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( cons_o @ ( nth_o @ Xs2 @ I ) @ ( drop_o @ ( suc @ I ) @ Xs2 ) )
        = ( drop_o @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8025_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs2 ) )
        = ( drop_nat @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8026_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs2: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( cons_int @ ( nth_int @ Xs2 @ I ) @ ( drop_int @ ( suc @ I ) @ Xs2 ) )
        = ( drop_int @ I @ Xs2 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_8027_in__set__drop__conv__nth,axiom,
    ! [X4: vEBT_VEBTi,N: nat,L: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ ( drop_VEBT_VEBTi @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L ) )
            & ( X4
              = ( nth_VEBT_VEBTi @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8028_in__set__drop__conv__nth,axiom,
    ! [X4: set_nat,N: nat,L: list_set_nat] :
      ( ( member_set_nat @ X4 @ ( set_set_nat2 @ ( drop_set_nat @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ L ) )
            & ( X4
              = ( nth_set_nat @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8029_in__set__drop__conv__nth,axiom,
    ! [X4: vEBT_VEBT,N: nat,L: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( drop_VEBT_VEBT @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L ) )
            & ( X4
              = ( nth_VEBT_VEBT @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8030_in__set__drop__conv__nth,axiom,
    ! [X4: real,N: nat,L: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ ( drop_real @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_size_list_real @ L ) )
            & ( X4
              = ( nth_real @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8031_in__set__drop__conv__nth,axiom,
    ! [X4: $o,N: nat,L: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ ( drop_o @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_size_list_o @ L ) )
            & ( X4
              = ( nth_o @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8032_in__set__drop__conv__nth,axiom,
    ! [X4: nat,N: nat,L: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ ( drop_nat @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
            & ( X4
              = ( nth_nat @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8033_in__set__drop__conv__nth,axiom,
    ! [X4: int,N: nat,L: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ ( drop_int @ N @ L ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ N @ I3 )
            & ( ord_less_nat @ I3 @ ( size_size_list_int @ L ) )
            & ( X4
              = ( nth_int @ L @ I3 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_8034_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_8035_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_8036_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_8037_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_8038_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P6: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8039_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8040_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8041_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8042_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P6: assn,M5: nat] : ( if_assn @ ( M5 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P6 @ ( power_power_assn @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8043_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P6: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_8044_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_8045_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_8046_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_8047_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_8048_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_8049_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_8050_sgn__power__injE,axiom,
    ! [A: real,N: nat,X4: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X4 )
     => ( ( X4
          = ( 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_8051_linear__plus__1__le__power,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X4 @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_8052_Bernoulli__inequality,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_8053_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8054_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8055_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8056_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8057_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_8058_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_8059_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_8060_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_8061_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_8062_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_8063_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_8064_drop__last__conv,axiom,
    ! [L: list_real] :
      ( ( L != nil_real )
     => ( ( drop_real @ ( minus_minus_nat @ ( size_size_list_real @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
        = ( cons_real @ ( last_real @ L ) @ nil_real ) ) ) ).

% drop_last_conv
thf(fact_8065_drop__last__conv,axiom,
    ! [L: list_o] :
      ( ( L != nil_o )
     => ( ( drop_o @ ( minus_minus_nat @ ( size_size_list_o @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
        = ( cons_o @ ( last_o @ L ) @ nil_o ) ) ) ).

% drop_last_conv
thf(fact_8066_drop__last__conv,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ( ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
        = ( cons_nat @ ( last_nat @ L ) @ nil_nat ) ) ) ).

% drop_last_conv
thf(fact_8067_drop__last__conv,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ( ( drop_int @ ( minus_minus_nat @ ( size_size_list_int @ L ) @ ( suc @ zero_zero_nat ) ) @ L )
        = ( cons_int @ ( last_int @ L ) @ nil_int ) ) ) ).

% drop_last_conv
thf(fact_8068_horner__sum__eq__sum,axiom,
    ( groups4473738407398356458Ti_rat
    = ( ^ [F4: vEBT_VEBTi > rat,A2: rat,Xs: list_VEBT_VEBTi] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_VEBT_VEBTi @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8069_horner__sum__eq__sum,axiom,
    ( groups6210023459537951729BT_rat
    = ( ^ [F4: vEBT_VEBT > rat,A2: rat,Xs: list_VEBT_VEBT] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_VEBT_VEBT @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8070_horner__sum__eq__sum,axiom,
    ( groups7414964958337966817al_rat
    = ( ^ [F4: real > rat,A2: rat,Xs: list_real] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_real @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8071_horner__sum__eq__sum,axiom,
    ( groups8483887719401441109_o_rat
    = ( ^ [F4: $o > rat,A2: rat,Xs: list_o] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_o @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8072_horner__sum__eq__sum,axiom,
    ( groups6853238114764508677at_rat
    = ( ^ [F4: nat > rat,A2: rat,Xs: list_nat] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_nat @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8073_horner__sum__eq__sum,axiom,
    ( groups7852591826665563233nt_rat
    = ( ^ [F4: int > rat,A2: rat,Xs: list_int] :
          ( groups2906978787729119204at_rat
          @ ^ [N4: nat] : ( times_times_rat @ ( F4 @ ( nth_int @ Xs @ N4 ) ) @ ( power_power_rat @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8074_horner__sum__eq__sum,axiom,
    ( groups5106377996975801918Ti_int
    = ( ^ [F4: vEBT_VEBTi > int,A2: int,Xs: list_VEBT_VEBTi] :
          ( groups3539618377306564664at_int
          @ ^ [N4: nat] : ( times_times_int @ ( F4 @ ( nth_VEBT_VEBTi @ Xs @ N4 ) ) @ ( power_power_int @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8075_horner__sum__eq__sum,axiom,
    ( groups6842663049115397189BT_int
    = ( ^ [F4: vEBT_VEBT > int,A2: int,Xs: list_VEBT_VEBT] :
          ( groups3539618377306564664at_int
          @ ^ [N4: nat] : ( times_times_int @ ( F4 @ ( nth_VEBT_VEBT @ Xs @ N4 ) ) @ ( power_power_int @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8076_horner__sum__eq__sum,axiom,
    ( groups8047604547915412277al_int
    = ( ^ [F4: real > int,A2: int,Xs: list_real] :
          ( groups3539618377306564664at_int
          @ ^ [N4: nat] : ( times_times_int @ ( F4 @ ( nth_real @ Xs @ N4 ) ) @ ( power_power_int @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8077_horner__sum__eq__sum,axiom,
    ( groups7485877704341954137at_int
    = ( ^ [F4: nat > int,A2: int,Xs: list_nat] :
          ( groups3539618377306564664at_int
          @ ^ [N4: nat] : ( times_times_int @ ( F4 @ ( nth_nat @ Xs @ N4 ) ) @ ( power_power_int @ A2 @ N4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8078_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ A )
        = ( append_VEBT_VEBTi @ ( take_VEBT_VEBTi @ I @ Xs2 ) @ ( cons_VEBT_VEBTi @ A @ ( drop_VEBT_VEBTi @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8079_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ A )
        = ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ I @ Xs2 ) @ ( cons_VEBT_VEBT @ A @ ( drop_VEBT_VEBT @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8080_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_real,A: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( list_update_real @ Xs2 @ I @ A )
        = ( append_real @ ( take_real @ I @ Xs2 ) @ ( cons_real @ A @ ( drop_real @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8081_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_o,A: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( list_update_o @ Xs2 @ I @ A )
        = ( append_o @ ( take_o @ I @ Xs2 ) @ ( cons_o @ A @ ( drop_o @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8082_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_nat,A: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( list_update_nat @ Xs2 @ I @ A )
        = ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8083_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs2: list_int,A: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( list_update_int @ Xs2 @ I @ A )
        = ( append_int @ ( take_int @ I @ Xs2 ) @ ( cons_int @ A @ ( drop_int @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_8084_concat__inth,axiom,
    ! [Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( append_VEBT_VEBTi @ Xs2 @ ( append_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ nil_VEBT_VEBTi ) @ Ys ) ) @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8085_concat__inth,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ Xs2 @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ nil_VEBT_VEBT ) @ Ys ) ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8086_concat__inth,axiom,
    ! [Xs2: list_real,X4: real,Ys: list_real] :
      ( ( nth_real @ ( append_real @ Xs2 @ ( append_real @ ( cons_real @ X4 @ nil_real ) @ Ys ) ) @ ( size_size_list_real @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8087_concat__inth,axiom,
    ! [Xs2: list_o,X4: $o,Ys: list_o] :
      ( ( nth_o @ ( append_o @ Xs2 @ ( append_o @ ( cons_o @ X4 @ nil_o ) @ Ys ) ) @ ( size_size_list_o @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8088_concat__inth,axiom,
    ! [Xs2: list_nat,X4: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs2 @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8089_concat__inth,axiom,
    ! [Xs2: list_int,X4: int,Ys: list_int] :
      ( ( nth_int @ ( append_int @ Xs2 @ ( append_int @ ( cons_int @ X4 @ nil_int ) @ Ys ) ) @ ( size_size_list_int @ Xs2 ) )
      = X4 ) ).

% concat_inth
thf(fact_8090_pos__n__replace,axiom,
    ! [N: nat,Xs2: list_real,Y: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ ( append_real @ ( take_real @ N @ Xs2 ) @ ( append_real @ ( cons_real @ Y @ nil_real ) @ ( drop_real @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).

% pos_n_replace
thf(fact_8091_pos__n__replace,axiom,
    ! [N: nat,Xs2: list_o,Y: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ ( append_o @ ( take_o @ N @ Xs2 ) @ ( append_o @ ( cons_o @ Y @ nil_o ) @ ( drop_o @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).

% pos_n_replace
thf(fact_8092_pos__n__replace,axiom,
    ! [N: nat,Xs2: list_nat,Y: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( append_nat @ ( cons_nat @ Y @ nil_nat ) @ ( drop_nat @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).

% pos_n_replace
thf(fact_8093_pos__n__replace,axiom,
    ! [N: nat,Xs2: list_int,Y: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ ( append_int @ ( take_int @ N @ Xs2 ) @ ( append_int @ ( cons_int @ Y @ nil_int ) @ ( drop_int @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).

% pos_n_replace
thf(fact_8094_append__eq__append__conv,axiom,
    ! [Xs2: list_real,Ys: list_real,Us: list_real,Vs: list_real] :
      ( ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Ys ) )
        | ( ( size_size_list_real @ Us )
          = ( size_size_list_real @ Vs ) ) )
     => ( ( ( append_real @ Xs2 @ Us )
          = ( append_real @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_8095_append__eq__append__conv,axiom,
    ! [Xs2: list_o,Ys: list_o,Us: list_o,Vs: list_o] :
      ( ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_o @ Ys ) )
        | ( ( size_size_list_o @ Us )
          = ( size_size_list_o @ Vs ) ) )
     => ( ( ( append_o @ Xs2 @ Us )
          = ( append_o @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_8096_append__eq__append__conv,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs2 @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_8097_append__eq__append__conv,axiom,
    ! [Xs2: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
      ( ( ( ( size_size_list_int @ Xs2 )
          = ( size_size_list_int @ Ys ) )
        | ( ( size_size_list_int @ Us )
          = ( size_size_list_int @ Vs ) ) )
     => ( ( ( append_int @ Xs2 @ Us )
          = ( append_int @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_8098_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A3 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_8099_sum_Oneutral__const,axiom,
    ! [A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A3 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_8100_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A3 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_8101_sum_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A3 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_8102_of__nat__sum,axiom,
    ! [F: complex > nat,A3: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_8103_of__nat__sum,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_8104_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_8105_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_8106_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_8107_abs__sum__abs,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ( abs_abs_real
        @ ( groups6591440286371151544t_real
          @ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
          @ A3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
        @ A3 ) ) ).

% abs_sum_abs
thf(fact_8108_abs__sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( abs_abs_int
        @ ( groups4538972089207619220nt_int
          @ ^ [A2: int] : ( abs_abs_int @ ( F @ A2 ) )
          @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [A2: int] : ( abs_abs_int @ ( F @ A2 ) )
        @ A3 ) ) ).

% abs_sum_abs
thf(fact_8109_nth__repl,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBTi,N: nat,X4: vEBT_VEBTi] :
      ( ( ord_less_nat @ M @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_VEBT_VEBTi @ ( append_VEBT_VEBTi @ ( take_VEBT_VEBTi @ N @ Xs2 ) @ ( append_VEBT_VEBTi @ ( cons_VEBT_VEBTi @ X4 @ nil_VEBT_VEBTi ) @ ( drop_VEBT_VEBTi @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_VEBT_VEBTi @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8110_nth__repl,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBT,N: nat,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ N @ Xs2 ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ X4 @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_VEBT_VEBT @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8111_nth__repl,axiom,
    ! [M: nat,Xs2: list_real,N: nat,X4: real] :
      ( ( ord_less_nat @ M @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_real @ ( append_real @ ( take_real @ N @ Xs2 ) @ ( append_real @ ( cons_real @ X4 @ nil_real ) @ ( drop_real @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_real @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8112_nth__repl,axiom,
    ! [M: nat,Xs2: list_o,N: nat,X4: $o] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_o @ ( append_o @ ( take_o @ N @ Xs2 ) @ ( append_o @ ( cons_o @ X4 @ nil_o ) @ ( drop_o @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_o @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8113_nth__repl,axiom,
    ! [M: nat,Xs2: list_nat,N: nat,X4: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_nat @ ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ ( drop_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_nat @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8114_nth__repl,axiom,
    ! [M: nat,Xs2: list_int,N: nat,X4: int] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( M != N )
         => ( ( nth_int @ ( append_int @ ( take_int @ N @ Xs2 ) @ ( append_int @ ( cons_int @ X4 @ nil_int ) @ ( drop_int @ ( plus_plus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) @ M )
            = ( nth_int @ Xs2 @ M ) ) ) ) ) ).

% nth_repl
thf(fact_8115_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_8116_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_8117_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8118_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8119_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_8120_sum_Oempty,axiom,
    ! [G: $o > complex] :
      ( ( groups5328290441151304332omplex @ G @ bot_bot_set_o )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_8121_sum_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups8691415230153176458o_real @ G @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_8122_sum_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8123_sum_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8124_sum_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_8125_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups2073611262835488442omplex @ G @ A3 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_8126_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups3049146728041665814omplex @ G @ A3 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_8127_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: code_integer > complex] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ( ( groups8024822376189712711omplex @ G @ A3 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_8128_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8129_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8130_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: code_integer > real] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1270011288395367621r_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8131_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8132_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8133_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8134_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8135_sum__eq__0__iff,axiom,
    ! [F5: set_int,F: int > nat] :
      ( ( finite_finite_int @ F5 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F5 )
          = zero_zero_nat )
        = ( ! [X: int] :
              ( ( member_int @ X @ F5 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8136_sum__eq__0__iff,axiom,
    ! [F5: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F5 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F5 )
          = zero_zero_nat )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ F5 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8137_sum__eq__0__iff,axiom,
    ! [F5: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ( groups7237345082560585321er_nat @ F @ F5 )
          = zero_zero_nat )
        = ( ! [X: code_integer] :
              ( ( member_Code_integer @ X @ F5 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8138_sum__eq__0__iff,axiom,
    ! [F5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F5 )
          = zero_zero_nat )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ F5 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8139_length__0__conv,axiom,
    ! [Xs2: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_real ) ) ).

% length_0_conv
thf(fact_8140_length__0__conv,axiom,
    ! [Xs2: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_o ) ) ).

% length_0_conv
thf(fact_8141_length__0__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_nat ) ) ).

% length_0_conv
thf(fact_8142_length__0__conv,axiom,
    ! [Xs2: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = zero_zero_nat )
      = ( Xs2 = nil_int ) ) ).

% length_0_conv
thf(fact_8143_set__empty2,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( bot_bo8194388402131092736T_VEBT
        = ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( Xs2 = nil_VEBT_VEBT ) ) ).

% set_empty2
thf(fact_8144_set__empty2,axiom,
    ! [Xs2: list_real] :
      ( ( bot_bot_set_real
        = ( set_real2 @ Xs2 ) )
      = ( Xs2 = nil_real ) ) ).

% set_empty2
thf(fact_8145_set__empty2,axiom,
    ! [Xs2: list_o] :
      ( ( bot_bot_set_o
        = ( set_o2 @ Xs2 ) )
      = ( Xs2 = nil_o ) ) ).

% set_empty2
thf(fact_8146_set__empty2,axiom,
    ! [Xs2: list_nat] :
      ( ( bot_bot_set_nat
        = ( set_nat2 @ Xs2 ) )
      = ( Xs2 = nil_nat ) ) ).

% set_empty2
thf(fact_8147_set__empty2,axiom,
    ! [Xs2: list_int] :
      ( ( bot_bot_set_int
        = ( set_int2 @ Xs2 ) )
      = ( Xs2 = nil_int ) ) ).

% set_empty2
thf(fact_8148_set__empty,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( ( set_VEBT_VEBT2 @ Xs2 )
        = bot_bo8194388402131092736T_VEBT )
      = ( Xs2 = nil_VEBT_VEBT ) ) ).

% set_empty
thf(fact_8149_set__empty,axiom,
    ! [Xs2: list_real] :
      ( ( ( set_real2 @ Xs2 )
        = bot_bot_set_real )
      = ( Xs2 = nil_real ) ) ).

% set_empty
thf(fact_8150_set__empty,axiom,
    ! [Xs2: list_o] :
      ( ( ( set_o2 @ Xs2 )
        = bot_bot_set_o )
      = ( Xs2 = nil_o ) ) ).

% set_empty
thf(fact_8151_set__empty,axiom,
    ! [Xs2: list_nat] :
      ( ( ( set_nat2 @ Xs2 )
        = bot_bot_set_nat )
      = ( Xs2 = nil_nat ) ) ).

% set_empty
thf(fact_8152_set__empty,axiom,
    ! [Xs2: list_int] :
      ( ( ( set_int2 @ Xs2 )
        = bot_bot_set_int )
      = ( Xs2 = nil_int ) ) ).

% set_empty
thf(fact_8153_length__append,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( size_size_list_real @ ( append_real @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_append
thf(fact_8154_length__append,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_size_list_o @ ( append_o @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_append
thf(fact_8155_length__append,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_8156_length__append,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( size_size_list_int @ ( append_int @ Xs2 @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_append
thf(fact_8157_take0,axiom,
    ( ( take_o @ zero_zero_nat )
    = ( ^ [Xs: list_o] : nil_o ) ) ).

% take0
thf(fact_8158_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs: list_nat] : nil_nat ) ) ).

% take0
thf(fact_8159_take0,axiom,
    ( ( take_int @ zero_zero_nat )
    = ( ^ [Xs: list_int] : nil_int ) ) ).

% take0
thf(fact_8160_take__eq__Nil,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ( take_o @ N @ Xs2 )
        = nil_o )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_o ) ) ) ).

% take_eq_Nil
thf(fact_8161_take__eq__Nil,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( take_nat @ N @ Xs2 )
        = nil_nat )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_nat ) ) ) ).

% take_eq_Nil
thf(fact_8162_take__eq__Nil,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ( take_int @ N @ Xs2 )
        = nil_int )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_int ) ) ) ).

% take_eq_Nil
thf(fact_8163_take__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( nil_o
        = ( take_o @ N @ Xs2 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_o ) ) ) ).

% take_eq_Nil2
thf(fact_8164_take__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( nil_nat
        = ( take_nat @ N @ Xs2 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_nat ) ) ) ).

% take_eq_Nil2
thf(fact_8165_take__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( nil_int
        = ( take_int @ N @ Xs2 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs2 = nil_int ) ) ) ).

% take_eq_Nil2
thf(fact_8166_replicate__empty,axiom,
    ! [N: nat,X4: nat] :
      ( ( ( replicate_nat @ N @ X4 )
        = nil_nat )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_8167_replicate__empty,axiom,
    ! [N: nat,X4: int] :
      ( ( ( replicate_int @ N @ X4 )
        = nil_int )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_8168_replicate__empty,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ N @ X4 )
        = nil_VEBT_VEBT )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_8169_replicate__empty,axiom,
    ! [N: nat,X4: $o] :
      ( ( ( replicate_o @ N @ X4 )
        = nil_o )
      = ( N = zero_zero_nat ) ) ).

% replicate_empty
thf(fact_8170_empty__replicate,axiom,
    ! [N: nat,X4: nat] :
      ( ( nil_nat
        = ( replicate_nat @ N @ X4 ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_8171_empty__replicate,axiom,
    ! [N: nat,X4: int] :
      ( ( nil_int
        = ( replicate_int @ N @ X4 ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_8172_empty__replicate,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( nil_VEBT_VEBT
        = ( replicate_VEBT_VEBT @ N @ X4 ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_8173_empty__replicate,axiom,
    ! [N: nat,X4: $o] :
      ( ( nil_o
        = ( replicate_o @ N @ X4 ) )
      = ( N = zero_zero_nat ) ) ).

% empty_replicate
thf(fact_8174_zip__append,axiom,
    ! [Xs2: list_real,Us: list_real,Ys: list_real,Vs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Us ) )
     => ( ( zip_real_real @ ( append_real @ Xs2 @ Ys ) @ ( append_real @ Us @ Vs ) )
        = ( append8511633455026463060l_real @ ( zip_real_real @ Xs2 @ Us ) @ ( zip_real_real @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8175_zip__append,axiom,
    ! [Xs2: list_real,Us: list_o,Ys: list_real,Vs: list_o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Us ) )
     => ( ( zip_real_o @ ( append_real @ Xs2 @ Ys ) @ ( append_o @ Us @ Vs ) )
        = ( append2727653632507881094real_o @ ( zip_real_o @ Xs2 @ Us ) @ ( zip_real_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8176_zip__append,axiom,
    ! [Xs2: list_real,Us: list_nat,Ys: list_real,Vs: list_nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_real_nat @ ( append_real @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append2703634050814417656al_nat @ ( zip_real_nat @ Xs2 @ Us ) @ ( zip_real_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8177_zip__append,axiom,
    ! [Xs2: list_real,Us: list_int,Ys: list_real,Vs: list_int] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Us ) )
     => ( ( zip_real_int @ ( append_real @ Xs2 @ Ys ) @ ( append_int @ Us @ Vs ) )
        = ( append7749155068159996756al_int @ ( zip_real_int @ Xs2 @ Us ) @ ( zip_real_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8178_zip__append,axiom,
    ! [Xs2: list_o,Us: list_real,Ys: list_o,Vs: list_real] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Us ) )
     => ( ( zip_o_real @ ( append_o @ Xs2 @ Ys ) @ ( append_real @ Us @ Vs ) )
        = ( append131710322270406936o_real @ ( zip_o_real @ Xs2 @ Us ) @ ( zip_o_real @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8179_zip__append,axiom,
    ! [Xs2: list_o,Us: list_o,Ys: list_o,Vs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Us ) )
     => ( ( zip_o_o @ ( append_o @ Xs2 @ Ys ) @ ( append_o @ Us @ Vs ) )
        = ( append2614242729457001410od_o_o @ ( zip_o_o @ Xs2 @ Us ) @ ( zip_o_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8180_zip__append,axiom,
    ! [Xs2: list_o,Us: list_nat,Ys: list_o,Vs: list_nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_o_nat @ ( append_o @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append7250250098557412540_o_nat @ ( zip_o_nat @ Xs2 @ Us ) @ ( zip_o_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8181_zip__append,axiom,
    ! [Xs2: list_o,Us: list_int,Ys: list_o,Vs: list_int] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_int @ Us ) )
     => ( ( zip_o_int @ ( append_o @ Xs2 @ Ys ) @ ( append_int @ Us @ Vs ) )
        = ( append3072399079048215832_o_int @ ( zip_o_int @ Xs2 @ Us ) @ ( zip_o_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8182_zip__append,axiom,
    ! [Xs2: list_nat,Us: list_real,Ys: list_nat,Vs: list_real] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_real @ Us ) )
     => ( ( zip_nat_real @ ( append_nat @ Xs2 @ Ys ) @ ( append_real @ Us @ Vs ) )
        = ( append6678681742291297912t_real @ ( zip_nat_real @ Xs2 @ Us ) @ ( zip_nat_real @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8183_zip__append,axiom,
    ! [Xs2: list_nat,Us: list_o,Ys: list_nat,Vs: list_o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_o @ Us ) )
     => ( ( zip_nat_o @ ( append_nat @ Xs2 @ Ys ) @ ( append_o @ Us @ Vs ) )
        = ( append1535412585758129762_nat_o @ ( zip_nat_o @ Xs2 @ Us ) @ ( zip_nat_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_8184_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord4252657396651189596t_real @ bot_bot_set_real )
    = nil_real ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_8185_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord3142498349692569832_set_o @ bot_bot_set_o )
    = nil_o ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_8186_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2612477271533052124et_int @ bot_bot_set_int )
    = nil_int ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_8187_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
    = nil_nat ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_8188_horner__sum__simps_I1_J,axiom,
    ! [F: nat > complex,A: complex] :
      ( ( groups404637655443745499omplex @ F @ A @ nil_nat )
      = zero_zero_complex ) ).

% horner_sum_simps(1)
thf(fact_8189_horner__sum__simps_I1_J,axiom,
    ! [F: int > complex,A: complex] :
      ( ( groups1380173120649922871omplex @ F @ A @ nil_int )
      = zero_zero_complex ) ).

% horner_sum_simps(1)
thf(fact_8190_horner__sum__simps_I1_J,axiom,
    ! [F: nat > real,A: real] :
      ( ( groups3482786445295563865t_real @ F @ A @ nil_nat )
      = zero_zero_real ) ).

% horner_sum_simps(1)
thf(fact_8191_horner__sum__simps_I1_J,axiom,
    ! [F: int > real,A: real] :
      ( ( groups5669708019988585653t_real @ F @ A @ nil_int )
      = zero_zero_real ) ).

% horner_sum_simps(1)
thf(fact_8192_horner__sum__simps_I1_J,axiom,
    ! [F: nat > rat,A: rat] :
      ( ( groups6853238114764508677at_rat @ F @ A @ nil_nat )
      = zero_zero_rat ) ).

% horner_sum_simps(1)
thf(fact_8193_horner__sum__simps_I1_J,axiom,
    ! [F: int > rat,A: rat] :
      ( ( groups7852591826665563233nt_rat @ F @ A @ nil_int )
      = zero_zero_rat ) ).

% horner_sum_simps(1)
thf(fact_8194_horner__sum__simps_I1_J,axiom,
    ! [F: nat > nat,A: nat] :
      ( ( groups7488368174851004413at_nat @ F @ A @ nil_nat )
      = zero_zero_nat ) ).

% horner_sum_simps(1)
thf(fact_8195_horner__sum__simps_I1_J,axiom,
    ! [F: int > nat,A: nat] :
      ( ( groups8487721886752058969nt_nat @ F @ A @ nil_int )
      = zero_zero_nat ) ).

% horner_sum_simps(1)
thf(fact_8196_horner__sum__simps_I1_J,axiom,
    ! [F: nat > int,A: int] :
      ( ( groups7485877704341954137at_int @ F @ A @ nil_nat )
      = zero_zero_int ) ).

% horner_sum_simps(1)
thf(fact_8197_horner__sum__simps_I1_J,axiom,
    ! [F: int > int,A: int] :
      ( ( groups8485231416243008693nt_int @ F @ A @ nil_int )
      = zero_zero_int ) ).

% horner_sum_simps(1)
thf(fact_8198_fun__upds__append__drop,axiom,
    ! [Xs2: list_real,Ys: list_real,M: real > option_real,Zs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_real_real @ M @ ( append_real @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_real_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8199_fun__upds__append__drop,axiom,
    ! [Xs2: list_real,Ys: list_o,M: real > option_o,Zs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_real_o @ M @ ( append_real @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_real_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8200_fun__upds__append__drop,axiom,
    ! [Xs2: list_real,Ys: list_nat,M: real > option_nat,Zs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_upds_real_nat @ M @ ( append_real @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_real_nat @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8201_fun__upds__append__drop,axiom,
    ! [Xs2: list_real,Ys: list_int,M: real > option_int,Zs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_upds_real_int @ M @ ( append_real @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_real_int @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8202_fun__upds__append__drop,axiom,
    ! [Xs2: list_o,Ys: list_real,M: $o > option_real,Zs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_o_real @ M @ ( append_o @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_o_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8203_fun__upds__append__drop,axiom,
    ! [Xs2: list_o,Ys: list_o,M: $o > option_o,Zs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_o_o @ M @ ( append_o @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_o_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8204_fun__upds__append__drop,axiom,
    ! [Xs2: list_o,Ys: list_nat,M: $o > option_nat,Zs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_upds_o_nat @ M @ ( append_o @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_o_nat @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8205_fun__upds__append__drop,axiom,
    ! [Xs2: list_o,Ys: list_int,M: $o > option_int,Zs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_upds_o_int @ M @ ( append_o @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_o_int @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8206_fun__upds__append__drop,axiom,
    ! [Xs2: list_nat,Ys: list_real,M: nat > option_real,Zs: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_nat_real @ M @ ( append_nat @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_nat_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8207_fun__upds__append__drop,axiom,
    ! [Xs2: list_nat,Ys: list_o,M: nat > option_o,Zs: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_nat_o @ M @ ( append_nat @ Xs2 @ Zs ) @ Ys )
        = ( map_upds_nat_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append_drop
thf(fact_8208_fun__upds__append2__drop,axiom,
    ! [Xs2: list_real,Ys: list_real,M: real > option_real,Zs: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_real_real @ M @ Xs2 @ ( append_real @ Ys @ Zs ) )
        = ( map_upds_real_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8209_fun__upds__append2__drop,axiom,
    ! [Xs2: list_real,Ys: list_o,M: real > option_o,Zs: list_o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_real_o @ M @ Xs2 @ ( append_o @ Ys @ Zs ) )
        = ( map_upds_real_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8210_fun__upds__append2__drop,axiom,
    ! [Xs2: list_real,Ys: list_nat,M: real > option_nat,Zs: list_nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_upds_real_nat @ M @ Xs2 @ ( append_nat @ Ys @ Zs ) )
        = ( map_upds_real_nat @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8211_fun__upds__append2__drop,axiom,
    ! [Xs2: list_real,Ys: list_int,M: real > option_int,Zs: list_int] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_upds_real_int @ M @ Xs2 @ ( append_int @ Ys @ Zs ) )
        = ( map_upds_real_int @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8212_fun__upds__append2__drop,axiom,
    ! [Xs2: list_o,Ys: list_real,M: $o > option_real,Zs: list_real] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_o_real @ M @ Xs2 @ ( append_real @ Ys @ Zs ) )
        = ( map_upds_o_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8213_fun__upds__append2__drop,axiom,
    ! [Xs2: list_o,Ys: list_o,M: $o > option_o,Zs: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_o_o @ M @ Xs2 @ ( append_o @ Ys @ Zs ) )
        = ( map_upds_o_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8214_fun__upds__append2__drop,axiom,
    ! [Xs2: list_o,Ys: list_nat,M: $o > option_nat,Zs: list_nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_upds_o_nat @ M @ Xs2 @ ( append_nat @ Ys @ Zs ) )
        = ( map_upds_o_nat @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8215_fun__upds__append2__drop,axiom,
    ! [Xs2: list_o,Ys: list_int,M: $o > option_int,Zs: list_int] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_upds_o_int @ M @ Xs2 @ ( append_int @ Ys @ Zs ) )
        = ( map_upds_o_int @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8216_fun__upds__append2__drop,axiom,
    ! [Xs2: list_nat,Ys: list_real,M: nat > option_real,Zs: list_real] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( map_upds_nat_real @ M @ Xs2 @ ( append_real @ Ys @ Zs ) )
        = ( map_upds_nat_real @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8217_fun__upds__append2__drop,axiom,
    ! [Xs2: list_nat,Ys: list_o,M: nat > option_o,Zs: list_o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_upds_nat_o @ M @ Xs2 @ ( append_o @ Ys @ Zs ) )
        = ( map_upds_nat_o @ M @ Xs2 @ Ys ) ) ) ).

% fun_upds_append2_drop
thf(fact_8218_list__assn__aux__simps_I2_J,axiom,
    ! [P: nat > nat > assn,L: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ L @ nil_nat )
      = ( pure_assn @ ( L = nil_nat ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_8219_list__assn__aux__simps_I2_J,axiom,
    ! [P: int > nat > assn,L: list_int] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ L @ nil_nat )
      = ( pure_assn @ ( L = nil_int ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_8220_list__assn__aux__simps_I2_J,axiom,
    ! [P: nat > int > assn,L: list_nat] :
      ( ( vEBT_L8298612041380073281at_int @ P @ L @ nil_int )
      = ( pure_assn @ ( L = nil_nat ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_8221_list__assn__aux__simps_I2_J,axiom,
    ! [P: int > int > assn,L: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ L @ nil_int )
      = ( pure_assn @ ( L = nil_int ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_8222_list__assn__aux__simps_I2_J,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,L: list_VEBT_VEBT] :
      ( ( vEBT_L6296928887356842470_VEBTi @ P @ L @ nil_VEBT_VEBTi )
      = ( pure_assn @ ( L = nil_VEBT_VEBT ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_8223_list__assn__aux__simps_I1_J,axiom,
    ! [P: nat > nat > assn,L4: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ nil_nat @ L4 )
      = ( pure_assn @ ( L4 = nil_nat ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_8224_list__assn__aux__simps_I1_J,axiom,
    ! [P: nat > int > assn,L4: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ nil_nat @ L4 )
      = ( pure_assn @ ( L4 = nil_int ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_8225_list__assn__aux__simps_I1_J,axiom,
    ! [P: int > nat > assn,L4: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ nil_int @ L4 )
      = ( pure_assn @ ( L4 = nil_nat ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_8226_list__assn__aux__simps_I1_J,axiom,
    ! [P: int > int > assn,L4: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ nil_int @ L4 )
      = ( pure_assn @ ( L4 = nil_int ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_8227_list__assn__aux__simps_I1_J,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,L4: list_VEBT_VEBTi] :
      ( ( vEBT_L6296928887356842470_VEBTi @ P @ nil_VEBT_VEBT @ L4 )
      = ( pure_assn @ ( L4 = nil_VEBT_VEBTi ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_8228_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8229_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8230_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8231_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8232_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > complex] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups8024822376189712711omplex
              @ ^ [K3: code_integer] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups8024822376189712711omplex
              @ ^ [K3: code_integer] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8233_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_8234_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_8235_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_8236_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_8237_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8238_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8239_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8240_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8241_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8242_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > complex] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups8024822376189712711omplex
              @ ^ [K3: code_integer] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups8024822376189712711omplex
              @ ^ [K3: code_integer] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8243_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_8244_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_8245_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_8246_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_8247_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8248_sum__abs,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_8249_sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_8250_length__greater__0__conv,axiom,
    ! [Xs2: list_real] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) )
      = ( Xs2 != nil_real ) ) ).

% length_greater_0_conv
thf(fact_8251_length__greater__0__conv,axiom,
    ! [Xs2: list_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) )
      = ( Xs2 != nil_o ) ) ).

% length_greater_0_conv
thf(fact_8252_length__greater__0__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
      = ( Xs2 != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_8253_length__greater__0__conv,axiom,
    ! [Xs2: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) )
      = ( Xs2 != nil_int ) ) ).

% length_greater_0_conv
thf(fact_8254_nth__append__first,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,L4: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( nth_VEBT_VEBTi @ ( append_VEBT_VEBTi @ L @ L4 ) @ I )
        = ( nth_VEBT_VEBTi @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8255_nth__append__first,axiom,
    ! [I: nat,L: list_VEBT_VEBT,L4: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ L @ L4 ) @ I )
        = ( nth_VEBT_VEBT @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8256_nth__append__first,axiom,
    ! [I: nat,L: list_real,L4: list_real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( nth_real @ ( append_real @ L @ L4 ) @ I )
        = ( nth_real @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8257_nth__append__first,axiom,
    ! [I: nat,L: list_o,L4: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( nth_o @ ( append_o @ L @ L4 ) @ I )
        = ( nth_o @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8258_nth__append__first,axiom,
    ! [I: nat,L: list_nat,L4: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( append_nat @ L @ L4 ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8259_nth__append__first,axiom,
    ! [I: nat,L: list_int,L4: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( append_int @ L @ L4 ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_append_first
thf(fact_8260_nth__append__length,axiom,
    ! [Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( nth_VEBT_VEBTi @ ( append_VEBT_VEBTi @ Xs2 @ ( cons_VEBT_VEBTi @ X4 @ Ys ) ) @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8261_nth__append__length,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ Xs2 @ ( cons_VEBT_VEBT @ X4 @ Ys ) ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8262_nth__append__length,axiom,
    ! [Xs2: list_real,X4: real,Ys: list_real] :
      ( ( nth_real @ ( append_real @ Xs2 @ ( cons_real @ X4 @ Ys ) ) @ ( size_size_list_real @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8263_nth__append__length,axiom,
    ! [Xs2: list_o,X4: $o,Ys: list_o] :
      ( ( nth_o @ ( append_o @ Xs2 @ ( cons_o @ X4 @ Ys ) ) @ ( size_size_list_o @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8264_nth__append__length,axiom,
    ! [Xs2: list_nat,X4: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8265_nth__append__length,axiom,
    ! [Xs2: list_int,X4: int,Ys: list_int] :
      ( ( nth_int @ ( append_int @ Xs2 @ ( cons_int @ X4 @ Ys ) ) @ ( size_size_list_int @ Xs2 ) )
      = X4 ) ).

% nth_append_length
thf(fact_8266_nth__append__length__plus,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,N: nat] :
      ( ( nth_VEBT_VEBTi @ ( append_VEBT_VEBTi @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ N ) )
      = ( nth_VEBT_VEBTi @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8267_nth__append__length__plus,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) )
      = ( nth_VEBT_VEBT @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8268_nth__append__length__plus,axiom,
    ! [Xs2: list_real,Ys: list_real,N: nat] :
      ( ( nth_real @ ( append_real @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_real @ Xs2 ) @ N ) )
      = ( nth_real @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8269_nth__append__length__plus,axiom,
    ! [Xs2: list_o,Ys: list_o,N: nat] :
      ( ( nth_o @ ( append_o @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ N ) )
      = ( nth_o @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8270_nth__append__length__plus,axiom,
    ! [Xs2: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8271_nth__append__length__plus,axiom,
    ! [Xs2: list_int,Ys: list_int,N: nat] :
      ( ( nth_int @ ( append_int @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ N ) )
      = ( nth_int @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_8272_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( nil_real
        = ( drop_real @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_8273_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( nil_o
        = ( drop_o @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_8274_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_8275_drop__eq__Nil2,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( nil_int
        = ( drop_int @ N @ Xs2 ) )
      = ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_8276_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ( drop_real @ N @ Xs2 )
        = nil_real )
      = ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_8277_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ( drop_o @ N @ Xs2 )
        = nil_o )
      = ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_8278_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( drop_nat @ N @ Xs2 )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_8279_drop__eq__Nil,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ( drop_int @ N @ Xs2 )
        = nil_int )
      = ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_8280_drop__all,axiom,
    ! [Xs2: list_real,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N )
     => ( ( drop_real @ N @ Xs2 )
        = nil_real ) ) ).

% drop_all
thf(fact_8281_drop__all,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N )
     => ( ( drop_o @ N @ Xs2 )
        = nil_o ) ) ).

% drop_all
thf(fact_8282_drop__all,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
     => ( ( drop_nat @ N @ Xs2 )
        = nil_nat ) ) ).

% drop_all
thf(fact_8283_drop__all,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N )
     => ( ( drop_int @ N @ Xs2 )
        = nil_int ) ) ).

% drop_all
thf(fact_8284_list__update__length,axiom,
    ! [Xs2: list_VEBT_VEBTi,X4: vEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( append_VEBT_VEBTi @ Xs2 @ ( cons_VEBT_VEBTi @ X4 @ Ys ) ) @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ Y )
      = ( append_VEBT_VEBTi @ Xs2 @ ( cons_VEBT_VEBTi @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8285_list__update__length,axiom,
    ! [Xs2: list_VEBT_VEBT,X4: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( append_VEBT_VEBT @ Xs2 @ ( cons_VEBT_VEBT @ X4 @ Ys ) ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ Y )
      = ( append_VEBT_VEBT @ Xs2 @ ( cons_VEBT_VEBT @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8286_list__update__length,axiom,
    ! [Xs2: list_real,X4: real,Ys: list_real,Y: real] :
      ( ( list_update_real @ ( append_real @ Xs2 @ ( cons_real @ X4 @ Ys ) ) @ ( size_size_list_real @ Xs2 ) @ Y )
      = ( append_real @ Xs2 @ ( cons_real @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8287_list__update__length,axiom,
    ! [Xs2: list_o,X4: $o,Ys: list_o,Y: $o] :
      ( ( list_update_o @ ( append_o @ Xs2 @ ( cons_o @ X4 @ Ys ) ) @ ( size_size_list_o @ Xs2 ) @ Y )
      = ( append_o @ Xs2 @ ( cons_o @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8288_list__update__length,axiom,
    ! [Xs2: list_nat,X4: nat,Ys: list_nat,Y: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) @ Y )
      = ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8289_list__update__length,axiom,
    ! [Xs2: list_int,X4: int,Ys: list_int,Y: int] :
      ( ( list_update_int @ ( append_int @ Xs2 @ ( cons_int @ X4 @ Ys ) ) @ ( size_size_list_int @ Xs2 ) @ Y )
      = ( append_int @ Xs2 @ ( cons_int @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_8290_take__append,axiom,
    ! [N: nat,Xs2: list_real,Ys: list_real] :
      ( ( take_real @ N @ ( append_real @ Xs2 @ Ys ) )
      = ( append_real @ ( take_real @ N @ Xs2 ) @ ( take_real @ ( minus_minus_nat @ N @ ( size_size_list_real @ Xs2 ) ) @ Ys ) ) ) ).

% take_append
thf(fact_8291_take__append,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_o] :
      ( ( take_o @ N @ ( append_o @ Xs2 @ Ys ) )
      = ( append_o @ ( take_o @ N @ Xs2 ) @ ( take_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs2 ) ) @ Ys ) ) ) ).

% take_append
thf(fact_8292_take__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( take_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% take_append
thf(fact_8293_take__append,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_int] :
      ( ( take_int @ N @ ( append_int @ Xs2 @ Ys ) )
      = ( append_int @ ( take_int @ N @ Xs2 ) @ ( take_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs2 ) ) @ Ys ) ) ) ).

% take_append
thf(fact_8294_drop__append,axiom,
    ! [N: nat,Xs2: list_real,Ys: list_real] :
      ( ( drop_real @ N @ ( append_real @ Xs2 @ Ys ) )
      = ( append_real @ ( drop_real @ N @ Xs2 ) @ ( drop_real @ ( minus_minus_nat @ N @ ( size_size_list_real @ Xs2 ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_8295_drop__append,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_o] :
      ( ( drop_o @ N @ ( append_o @ Xs2 @ Ys ) )
      = ( append_o @ ( drop_o @ N @ Xs2 ) @ ( drop_o @ ( minus_minus_nat @ N @ ( size_size_list_o @ Xs2 ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_8296_drop__append,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( drop_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_8297_drop__append,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_int] :
      ( ( drop_int @ N @ ( append_int @ Xs2 @ Ys ) )
      = ( append_int @ ( drop_int @ N @ Xs2 ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs2 ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_8298_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ( linord2324613341767563021nteger @ A3 )
          = nil_Code_integer )
        = ( A3 = bot_bo3990330152332043303nteger ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_8299_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( linord4252657396651189596t_real @ A3 )
          = nil_real )
        = ( A3 = bot_bot_set_real ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_8300_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( linord3142498349692569832_set_o @ A3 )
          = nil_o )
        = ( A3 = bot_bot_set_o ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_8301_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( linord2612477271533052124et_int @ A3 )
          = nil_int )
        = ( A3 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_8302_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( linord2614967742042102400et_nat @ A3 )
          = nil_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_8303_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_real,P: real > real > assn,L1: list_real,L13: list_real] :
      ( ( ( size_size_list_real @ L22 )
        = ( size_size_list_real @ L24 ) )
     => ( ( vEBT_L1930518968523514909l_real @ P @ ( append_real @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1930518968523514909l_real @ P @ L1 @ L13 ) @ ( vEBT_L1930518968523514909l_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8304_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_o,P: real > $o > assn,L1: list_real,L13: list_o] :
      ( ( ( size_size_list_real @ L22 )
        = ( size_size_list_o @ L24 ) )
     => ( ( vEBT_L6234343332106409831real_o @ P @ ( append_real @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L6234343332106409831real_o @ P @ L1 @ L13 ) @ ( vEBT_L6234343332106409831real_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8305_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_nat,P: real > nat > assn,L1: list_real,L13: list_nat] :
      ( ( ( size_size_list_real @ L22 )
        = ( size_size_list_nat @ L24 ) )
     => ( ( vEBT_L1446010312343316929al_nat @ P @ ( append_real @ L1 @ L22 ) @ ( append_nat @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ P @ L1 @ L13 ) @ ( vEBT_L1446010312343316929al_nat @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8306_list__assn__aux__append2,axiom,
    ! [L22: list_real,L24: list_int,P: real > int > assn,L1: list_real,L13: list_int] :
      ( ( ( size_size_list_real @ L22 )
        = ( size_size_list_int @ L24 ) )
     => ( ( vEBT_L1443519841834266653al_int @ P @ ( append_real @ L1 @ L22 ) @ ( append_int @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1443519841834266653al_int @ P @ L1 @ L13 ) @ ( vEBT_L1443519841834266653al_int @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8307_list__assn__aux__append2,axiom,
    ! [L22: list_o,L24: list_real,P: $o > real > assn,L1: list_o,L13: list_real] :
      ( ( ( size_size_list_o @ L22 )
        = ( size_size_list_real @ L24 ) )
     => ( ( vEBT_L4725278957065240257o_real @ P @ ( append_o @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4725278957065240257o_real @ P @ L1 @ L13 ) @ ( vEBT_L4725278957065240257o_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8308_list__assn__aux__append2,axiom,
    ! [L22: list_o,L24: list_o,P: $o > $o > assn,L1: list_o,L13: list_o] :
      ( ( ( size_size_list_o @ L22 )
        = ( size_size_list_o @ L24 ) )
     => ( ( vEBT_L7363604446928714179sn_o_o @ P @ ( append_o @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L7363604446928714179sn_o_o @ P @ L1 @ L13 ) @ ( vEBT_L7363604446928714179sn_o_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8309_list__assn__aux__append2,axiom,
    ! [L22: list_o,L24: list_nat,P: $o > nat > assn,L1: list_o,L13: list_nat] :
      ( ( ( size_size_list_o @ L22 )
        = ( size_size_list_nat @ L24 ) )
     => ( ( vEBT_L4785011123346445925_o_nat @ P @ ( append_o @ L1 @ L22 ) @ ( append_nat @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4785011123346445925_o_nat @ P @ L1 @ L13 ) @ ( vEBT_L4785011123346445925_o_nat @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8310_list__assn__aux__append2,axiom,
    ! [L22: list_o,L24: list_int,P: $o > int > assn,L1: list_o,L13: list_int] :
      ( ( ( size_size_list_o @ L22 )
        = ( size_size_list_int @ L24 ) )
     => ( ( vEBT_L4782520652837395649_o_int @ P @ ( append_o @ L1 @ L22 ) @ ( append_int @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4782520652837395649_o_int @ P @ L1 @ L13 ) @ ( vEBT_L4782520652837395649_o_int @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8311_list__assn__aux__append2,axiom,
    ! [L22: list_nat,L24: list_real,P: nat > real > assn,L1: list_nat,L13: list_real] :
      ( ( ( size_size_list_nat @ L22 )
        = ( size_size_list_real @ L24 ) )
     => ( ( vEBT_L6102073776069194049t_real @ P @ ( append_nat @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L6102073776069194049t_real @ P @ L1 @ L13 ) @ ( vEBT_L6102073776069194049t_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8312_list__assn__aux__append2,axiom,
    ! [L22: list_nat,L24: list_o,P: nat > $o > assn,L1: list_nat,L13: list_o] :
      ( ( ( size_size_list_nat @ L22 )
        = ( size_size_list_o @ L24 ) )
     => ( ( vEBT_L7887682484454631235_nat_o @ P @ ( append_nat @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L7887682484454631235_nat_o @ P @ L1 @ L13 ) @ ( vEBT_L7887682484454631235_nat_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_8313_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_real,P: real > real > assn,L22: list_real,L24: list_real] :
      ( ( ( size_size_list_real @ L1 )
        = ( size_size_list_real @ L13 ) )
     => ( ( vEBT_L1930518968523514909l_real @ P @ ( append_real @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1930518968523514909l_real @ P @ L1 @ L13 ) @ ( vEBT_L1930518968523514909l_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8314_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_o,P: real > $o > assn,L22: list_real,L24: list_o] :
      ( ( ( size_size_list_real @ L1 )
        = ( size_size_list_o @ L13 ) )
     => ( ( vEBT_L6234343332106409831real_o @ P @ ( append_real @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L6234343332106409831real_o @ P @ L1 @ L13 ) @ ( vEBT_L6234343332106409831real_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8315_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_nat,P: real > nat > assn,L22: list_real,L24: list_nat] :
      ( ( ( size_size_list_real @ L1 )
        = ( size_size_list_nat @ L13 ) )
     => ( ( vEBT_L1446010312343316929al_nat @ P @ ( append_real @ L1 @ L22 ) @ ( append_nat @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ P @ L1 @ L13 ) @ ( vEBT_L1446010312343316929al_nat @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8316_list__assn__aux__append,axiom,
    ! [L1: list_real,L13: list_int,P: real > int > assn,L22: list_real,L24: list_int] :
      ( ( ( size_size_list_real @ L1 )
        = ( size_size_list_int @ L13 ) )
     => ( ( vEBT_L1443519841834266653al_int @ P @ ( append_real @ L1 @ L22 ) @ ( append_int @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L1443519841834266653al_int @ P @ L1 @ L13 ) @ ( vEBT_L1443519841834266653al_int @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8317_list__assn__aux__append,axiom,
    ! [L1: list_o,L13: list_real,P: $o > real > assn,L22: list_o,L24: list_real] :
      ( ( ( size_size_list_o @ L1 )
        = ( size_size_list_real @ L13 ) )
     => ( ( vEBT_L4725278957065240257o_real @ P @ ( append_o @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4725278957065240257o_real @ P @ L1 @ L13 ) @ ( vEBT_L4725278957065240257o_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8318_list__assn__aux__append,axiom,
    ! [L1: list_o,L13: list_o,P: $o > $o > assn,L22: list_o,L24: list_o] :
      ( ( ( size_size_list_o @ L1 )
        = ( size_size_list_o @ L13 ) )
     => ( ( vEBT_L7363604446928714179sn_o_o @ P @ ( append_o @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L7363604446928714179sn_o_o @ P @ L1 @ L13 ) @ ( vEBT_L7363604446928714179sn_o_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8319_list__assn__aux__append,axiom,
    ! [L1: list_o,L13: list_nat,P: $o > nat > assn,L22: list_o,L24: list_nat] :
      ( ( ( size_size_list_o @ L1 )
        = ( size_size_list_nat @ L13 ) )
     => ( ( vEBT_L4785011123346445925_o_nat @ P @ ( append_o @ L1 @ L22 ) @ ( append_nat @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4785011123346445925_o_nat @ P @ L1 @ L13 ) @ ( vEBT_L4785011123346445925_o_nat @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8320_list__assn__aux__append,axiom,
    ! [L1: list_o,L13: list_int,P: $o > int > assn,L22: list_o,L24: list_int] :
      ( ( ( size_size_list_o @ L1 )
        = ( size_size_list_int @ L13 ) )
     => ( ( vEBT_L4782520652837395649_o_int @ P @ ( append_o @ L1 @ L22 ) @ ( append_int @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L4782520652837395649_o_int @ P @ L1 @ L13 ) @ ( vEBT_L4782520652837395649_o_int @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8321_list__assn__aux__append,axiom,
    ! [L1: list_nat,L13: list_real,P: nat > real > assn,L22: list_nat,L24: list_real] :
      ( ( ( size_size_list_nat @ L1 )
        = ( size_size_list_real @ L13 ) )
     => ( ( vEBT_L6102073776069194049t_real @ P @ ( append_nat @ L1 @ L22 ) @ ( append_real @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L6102073776069194049t_real @ P @ L1 @ L13 ) @ ( vEBT_L6102073776069194049t_real @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8322_list__assn__aux__append,axiom,
    ! [L1: list_nat,L13: list_o,P: nat > $o > assn,L22: list_nat,L24: list_o] :
      ( ( ( size_size_list_nat @ L1 )
        = ( size_size_list_o @ L13 ) )
     => ( ( vEBT_L7887682484454631235_nat_o @ P @ ( append_nat @ L1 @ L22 ) @ ( append_o @ L13 @ L24 ) )
        = ( times_times_assn @ ( vEBT_L7887682484454631235_nat_o @ P @ L1 @ L13 ) @ ( vEBT_L7887682484454631235_nat_o @ P @ L22 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_8323_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_8324_sum__abs__ge__zero,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_8325_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_8326_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = ( cons_list_int @ nil_int @ nil_list_int ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = nil_list_int ) ) ) ).

% n_lists_Nil
thf(fact_8327_length__ge__1__conv,axiom,
    ! [L: list_real] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_real @ L ) )
      = ( L != nil_real ) ) ).

% length_ge_1_conv
thf(fact_8328_length__ge__1__conv,axiom,
    ! [L: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_o @ L ) )
      = ( L != nil_o ) ) ).

% length_ge_1_conv
thf(fact_8329_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ L ) )
      = ( L != nil_nat ) ) ).

% length_ge_1_conv
thf(fact_8330_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_int @ L ) )
      = ( L != nil_int ) ) ).

% length_ge_1_conv
thf(fact_8331_concat__map__singleton,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( concat_real
        @ ( map_VE755361874264402988t_real
          @ ^ [X: vEBT_VEBT] : ( cons_real @ ( F @ X ) @ nil_real )
          @ Xs2 ) )
      = ( map_VEBT_VEBT_real @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_8332_concat__map__singleton,axiom,
    ! [F: product_prod_o_o > $o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( concat_o
        @ ( map_Pr8327783502415007611list_o
          @ ^ [X: product_prod_o_o] : ( cons_o @ ( F @ X ) @ nil_o )
          @ Xs2 ) )
      = ( map_Pr7541730621154948341_o_o_o @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_8333_concat__map__singleton,axiom,
    ! [F: nat > $o,Xs2: list_nat] :
      ( ( concat_o
        @ ( map_nat_list_o
          @ ^ [X: nat] : ( cons_o @ ( F @ X ) @ nil_o )
          @ Xs2 ) )
      = ( map_nat_o @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_8334_concat__map__singleton,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT] :
      ( ( concat_nat
        @ ( map_VE3614494202555519440st_nat
          @ ^ [X: vEBT_VEBT] : ( cons_nat @ ( F @ X ) @ nil_nat )
          @ Xs2 ) )
      = ( map_VEBT_VEBT_nat @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_8335_concat__map__singleton,axiom,
    ! [F: nat > nat,Xs2: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X: nat] : ( cons_nat @ ( F @ X ) @ nil_nat )
          @ Xs2 ) )
      = ( map_nat_nat @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_8336_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8337_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_8338_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_8339_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_8340_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_8341_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8342_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A3 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8343_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A3 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8344_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBTi,M: real > option_VEBT_VEBTi,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ( map_up4277394084060657369_VEBTi @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up5845057532826650672_VEBTi @ ( map_up4277394084060657369_VEBTi @ M @ Xs2 @ Ys ) @ X4 @ ( some_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8345_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT,M: real > option_VEBT_VEBT,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( map_up3774473461586936848T_VEBT @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up3388931106328796303T_VEBT @ ( map_up3774473461586936848T_VEBT @ M @ Xs2 @ Ys ) @ X4 @ ( some_VEBT_VEBT @ ( nth_VEBT_VEBT @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8346_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_num,M: real > option_num,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_num @ Ys ) )
     => ( ( map_upds_real_num @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up7385324149885497365on_num @ ( map_upds_real_num @ M @ Xs2 @ Ys ) @ X4 @ ( some_num @ ( nth_num @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8347_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_real,M: real > option_real,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) )
     => ( ( map_upds_real_real @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up7779911115344551847n_real @ ( map_upds_real_real @ M @ Xs2 @ Ys ) @ X4 @ ( some_real @ ( nth_real @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8348_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_o,M: real > option_o,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_o @ Ys ) )
     => ( ( map_upds_real_o @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up7055717989807170547tion_o @ ( map_upds_real_o @ M @ Xs2 @ Ys ) @ X4 @ ( some_o @ ( nth_o @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8349_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_nat,M: real > option_nat,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
     => ( ( map_upds_real_nat @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up6677080212936659659on_nat @ ( map_upds_real_nat @ M @ Xs2 @ Ys ) @ X4 @ ( some_nat @ ( nth_nat @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8350_map__upds__append1,axiom,
    ! [Xs2: list_real,Ys: list_int,M: real > option_int,X4: real] :
      ( ( ord_less_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_int @ Ys ) )
     => ( ( map_upds_real_int @ M @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) @ Ys )
        = ( fun_up2499229193427462951on_int @ ( map_upds_real_int @ M @ Xs2 @ Ys ) @ X4 @ ( some_int @ ( nth_int @ Ys @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8351_map__upds__append1,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBTi,M: $o > option_VEBT_VEBTi,X4: $o] :
      ( ( ord_less_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ( map_up3026452068722249431_VEBTi @ M @ ( append_o @ Xs2 @ ( cons_o @ X4 @ nil_o ) ) @ Ys )
        = ( fun_up3586965760925357280_VEBTi @ ( map_up3026452068722249431_VEBTi @ M @ Xs2 @ Ys ) @ X4 @ ( some_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Ys @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8352_map__upds__append1,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT,M: $o > option_VEBT_VEBT,X4: $o] :
      ( ( ord_less_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( map_upds_o_VEBT_VEBT @ M @ ( append_o @ Xs2 @ ( cons_o @ X4 @ nil_o ) ) @ Ys )
        = ( fun_up7750356280259972063T_VEBT @ ( map_upds_o_VEBT_VEBT @ M @ Xs2 @ Ys ) @ X4 @ ( some_VEBT_VEBT @ ( nth_VEBT_VEBT @ Ys @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8353_map__upds__append1,axiom,
    ! [Xs2: list_o,Ys: list_num,M: $o > option_num,X4: $o] :
      ( ( ord_less_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_num @ Ys ) )
     => ( ( map_upds_o_num @ M @ ( append_o @ Xs2 @ ( cons_o @ X4 @ nil_o ) ) @ Ys )
        = ( fun_upd_o_option_num @ ( map_upds_o_num @ M @ Xs2 @ Ys ) @ X4 @ ( some_num @ ( nth_num @ Ys @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).

% map_upds_append1
thf(fact_8354_same__length__different,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Ys ) )
       => ? [Pre: list_real,X3: real,Xs5: list_real,Y3: real,Ys4: list_real] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_real @ Pre @ ( append_real @ ( cons_real @ X3 @ nil_real ) @ Xs5 ) ) )
            & ( Ys
              = ( append_real @ Pre @ ( append_real @ ( cons_real @ Y3 @ nil_real ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_8355_same__length__different,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_o @ Ys ) )
       => ? [Pre: list_o,X3: $o,Xs5: list_o,Y3: $o,Ys4: list_o] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_o @ Pre @ ( append_o @ ( cons_o @ X3 @ nil_o ) @ Xs5 ) ) )
            & ( Ys
              = ( append_o @ Pre @ ( append_o @ ( cons_o @ Y3 @ nil_o ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_8356_same__length__different,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y3: nat,Ys4: list_nat] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_8357_same__length__different,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size_list_int @ Xs2 )
          = ( size_size_list_int @ Ys ) )
       => ? [Pre: list_int,X3: int,Xs5: list_int,Y3: int,Ys4: list_int] :
            ( ( X3 != Y3 )
            & ( Xs2
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ X3 @ nil_int ) @ Xs5 ) ) )
            & ( Ys
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_8358_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B4: set_nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B4 )
        @ A3 )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A3 )
        @ B4 ) ) ).

% sum.swap
thf(fact_8359_sum_Oswap,axiom,
    ! [G: complex > complex > complex,B4: set_complex,A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B4 )
        @ A3 )
      = ( groups7754918857620584856omplex
        @ ^ [J3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [I3: complex] : ( G @ I3 @ J3 )
            @ A3 )
        @ B4 ) ) ).

% sum.swap
thf(fact_8360_sum_Oswap,axiom,
    ! [G: nat > nat > real,B4: set_nat,A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B4 )
        @ A3 )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A3 )
        @ B4 ) ) ).

% sum.swap
thf(fact_8361_sum_Oswap,axiom,
    ! [G: int > int > int,B4: set_int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B4 )
        @ A3 )
      = ( groups4538972089207619220nt_int
        @ ^ [J3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I3: int] : ( G @ I3 @ J3 )
            @ A3 )
        @ B4 ) ) ).

% sum.swap
thf(fact_8362_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A3: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G @ A3 )
       != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8363_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > complex,A3: set_VEBT_VEBT] :
      ( ( ( groups1794756597179926696omplex @ G @ A3 )
       != zero_zero_complex )
     => ~ ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8364_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A3: set_real] :
      ( ( ( groups5754745047067104278omplex @ G @ A3 )
       != zero_zero_complex )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8365_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A3: set_int] :
      ( ( ( groups3049146728041665814omplex @ G @ A3 )
       != zero_zero_complex )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8366_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A3: set_VEBT_VEBT] :
      ( ( ( groups2240296850493347238T_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8367_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A3: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8368_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A3: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8369_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A3: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8370_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > rat,A3: set_VEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8371_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A3: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8372_sum_Oneutral,axiom,
    ! [A3: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( G @ X3 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A3 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_8373_sum_Oneutral,axiom,
    ! [A3: set_complex,G: complex > complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A3 )
         => ( ( G @ X3 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A3 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_8374_sum_Oneutral,axiom,
    ! [A3: set_nat,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( G @ X3 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_8375_sum_Oneutral,axiom,
    ! [A3: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( G @ X3 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A3 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_8376_map__by__foldl,axiom,
    ! [F: vEBT_VEBT > real,L: list_VEBT_VEBT] :
      ( ( foldl_4796148479939130645T_VEBT
        @ ^ [L3: list_real,X: vEBT_VEBT] : ( append_real @ L3 @ ( cons_real @ ( F @ X ) @ nil_real ) )
        @ nil_real
        @ L )
      = ( map_VEBT_VEBT_real @ F @ L ) ) ).

% map_by_foldl
thf(fact_8377_map__by__foldl,axiom,
    ! [F: product_prod_o_o > $o,L: list_P4002435161011370285od_o_o] :
      ( ( foldl_6187942973324179882od_o_o
        @ ^ [L3: list_o,X: product_prod_o_o] : ( append_o @ L3 @ ( cons_o @ ( F @ X ) @ nil_o ) )
        @ nil_o
        @ L )
      = ( map_Pr7541730621154948341_o_o_o @ F @ L ) ) ).

% map_by_foldl
thf(fact_8378_map__by__foldl,axiom,
    ! [F: nat > $o,L: list_nat] :
      ( ( foldl_list_o_nat
        @ ^ [L3: list_o,X: nat] : ( append_o @ L3 @ ( cons_o @ ( F @ X ) @ nil_o ) )
        @ nil_o
        @ L )
      = ( map_nat_o @ F @ L ) ) ).

% map_by_foldl
thf(fact_8379_map__by__foldl,axiom,
    ! [F: vEBT_VEBT > nat,L: list_VEBT_VEBT] :
      ( ( foldl_7309499613077172081T_VEBT
        @ ^ [L3: list_nat,X: vEBT_VEBT] : ( append_nat @ L3 @ ( cons_nat @ ( F @ X ) @ nil_nat ) )
        @ nil_nat
        @ L )
      = ( map_VEBT_VEBT_nat @ F @ L ) ) ).

% map_by_foldl
thf(fact_8380_map__by__foldl,axiom,
    ! [F: nat > nat,L: list_nat] :
      ( ( foldl_list_nat_nat
        @ ^ [L3: list_nat,X: nat] : ( append_nat @ L3 @ ( cons_nat @ ( F @ X ) @ nil_nat ) )
        @ nil_nat
        @ L )
      = ( map_nat_nat @ F @ L ) ) ).

% map_by_foldl
thf(fact_8381_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc1828647624359046049st_nat] :
      ( ! [Ys5: list_nat] :
          ( X4
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) )
     => ~ ! [X3: nat,Xs3: list_nat,Ys5: list_nat] :
            ( X4
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Ys5 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_8382_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc1186641810826059865st_int] :
      ( ! [Ys5: list_int] :
          ( X4
         != ( produc364263696895485585st_int @ nil_int @ Ys5 ) )
     => ~ ! [X3: int,Xs3: list_int,Ys5: list_int] :
            ( X4
           != ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ Ys5 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_8383_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8384_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ K5 ) @ ( groups136491112297645522BT_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8385_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8386_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8387_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ K5 ) @ ( groups771621172384141258BT_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8388_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8389_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8390_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8391_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups769130701875090982BT_int @ F @ K5 ) @ ( groups769130701875090982BT_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8392_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8393_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_8394_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A3 ) @ ( groups7754918857620584856omplex @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_8395_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A3 ) @ ( groups6591440286371151544t_real @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_8396_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A3 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A3 ) @ ( groups4538972089207619220nt_int @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_8397_sum__product,axiom,
    ! [F: nat > nat,A3: set_nat,G: nat > nat,B4: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A3 ) ) ).

% sum_product
thf(fact_8398_sum__product,axiom,
    ! [F: complex > complex,A3: set_complex,G: complex > complex,B4: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ ( groups7754918857620584856omplex @ G @ B4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A3 ) ) ).

% sum_product
thf(fact_8399_sum__product,axiom,
    ! [F: nat > real,A3: set_nat,G: nat > real,B4: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ ( groups6591440286371151544t_real @ G @ B4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A3 ) ) ).

% sum_product
thf(fact_8400_sum__product,axiom,
    ! [F: int > int,A3: set_int,G: int > int,B4: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ G @ B4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A3 ) ) ).

% sum_product
thf(fact_8401_sum__distrib__right,axiom,
    ! [F: nat > nat,A3: set_nat,R3: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ R3 )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_8402_sum__distrib__right,axiom,
    ! [F: complex > complex,A3: set_complex,R3: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ R3 )
      = ( groups7754918857620584856omplex
        @ ^ [N4: complex] : ( times_times_complex @ ( F @ N4 ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_8403_sum__distrib__right,axiom,
    ! [F: nat > real,A3: set_nat,R3: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ R3 )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_8404_sum__distrib__right,axiom,
    ! [F: int > int,A3: set_int,R3: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ R3 )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ ( F @ N4 ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_8405_sum__distrib__left,axiom,
    ! [R3: nat,F: nat > nat,A3: set_nat] :
      ( ( times_times_nat @ R3 @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ R3 @ ( F @ N4 ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_8406_sum__distrib__left,axiom,
    ! [R3: complex,F: complex > complex,A3: set_complex] :
      ( ( times_times_complex @ R3 @ ( groups7754918857620584856omplex @ F @ A3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N4: complex] : ( times_times_complex @ R3 @ ( F @ N4 ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_8407_sum__distrib__left,axiom,
    ! [R3: real,F: nat > real,A3: set_nat] :
      ( ( times_times_real @ R3 @ ( groups6591440286371151544t_real @ F @ A3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ R3 @ ( F @ N4 ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_8408_sum__distrib__left,axiom,
    ! [R3: int,F: int > int,A3: set_int] :
      ( ( times_times_int @ R3 @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ R3 @ ( F @ N4 ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_8409_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
        @ A3 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ ( groups7754918857620584856omplex @ G @ A3 ) ) ) ).

% sum_subtractf
thf(fact_8410_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
        @ A3 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ ( groups6591440286371151544t_real @ G @ A3 ) ) ) ).

% sum_subtractf
thf(fact_8411_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
        @ A3 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ G @ A3 ) ) ) ).

% sum_subtractf
thf(fact_8412_sum__negf,axiom,
    ! [F: complex > complex,A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( uminus1482373934393186551omplex @ ( F @ X ) )
        @ A3 )
      = ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A3 ) ) ) ).

% sum_negf
thf(fact_8413_sum__negf,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( uminus_uminus_real @ ( F @ X ) )
        @ A3 )
      = ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).

% sum_negf
thf(fact_8414_sum__negf,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( uminus_uminus_int @ ( F @ X ) )
        @ A3 )
      = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) ) ) ).

% sum_negf
thf(fact_8415_sum_Oswap__restrict,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_nat,G: vEBT_VEBT > nat > nat,R: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups771621172384141258BT_nat
            @ ^ [X: vEBT_VEBT] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups771621172384141258BT_nat
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y4 )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8416_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X: real] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X: real] : ( G @ X @ Y4 )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8417_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B4: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X: int] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X: int] : ( G @ X @ Y4 )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8418_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X: complex] : ( G @ X @ Y4 )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8419_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B4: set_nat,G: code_integer > nat > nat,R: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X: code_integer] : ( G @ X @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8420_sum_Oswap__restrict,axiom,
    ! [A3: set_VEBT_VEBT,B4: set_complex,G: vEBT_VEBT > complex > complex,R: vEBT_VEBT > complex > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups1794756597179926696omplex
            @ ^ [X: vEBT_VEBT] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups1794756597179926696omplex
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y4 )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8421_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_complex,G: real > complex > complex,R: real > complex > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X: real] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X: real] : ( G @ X @ Y4 )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8422_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X: nat] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X: nat] : ( G @ X @ Y4 )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8423_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B4: set_complex,G: int > complex > complex,R: int > complex > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X: int] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X: int] : ( G @ X @ Y4 )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8424_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B4: set_complex,G: code_integer > complex > complex,R: code_integer > complex > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups8024822376189712711omplex
            @ ^ [X: code_integer] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y4: complex] :
                      ( ( member_complex @ Y4 @ B4 )
                      & ( R @ X @ Y4 ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y4: complex] :
                ( groups8024822376189712711omplex
                @ ^ [X: code_integer] : ( G @ X @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A3 )
                      & ( R @ X @ Y4 ) ) ) )
            @ B4 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8425_length__Suc__rev__conv,axiom,
    ! [Xs2: list_real,N: nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Ys3: list_real,Y4: real] :
            ( ( Xs2
              = ( append_real @ Ys3 @ ( cons_real @ Y4 @ nil_real ) ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_8426_length__Suc__rev__conv,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Ys3: list_o,Y4: $o] :
            ( ( Xs2
              = ( append_o @ Ys3 @ ( cons_o @ Y4 @ nil_o ) ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_8427_length__Suc__rev__conv,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Ys3: list_nat,Y4: nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_8428_length__Suc__rev__conv,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Ys3: list_int,Y4: int] :
            ( ( Xs2
              = ( append_int @ Ys3 @ ( cons_int @ Y4 @ nil_int ) ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_8429_length__Suc__conv__rev,axiom,
    ! [Xs2: list_real,N: nat] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: real,Ys3: list_real] :
            ( ( Xs2
              = ( append_real @ Ys3 @ ( cons_real @ Y4 @ nil_real ) ) )
            & ( ( size_size_list_real @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_8430_length__Suc__conv__rev,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( append_o @ Ys3 @ ( cons_o @ Y4 @ nil_o ) ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_8431_length__Suc__conv__rev,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_8432_length__Suc__conv__rev,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs2
              = ( append_int @ Ys3 @ ( cons_int @ Y4 @ nil_int ) ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_8433_length__append__singleton,axiom,
    ! [Xs2: list_real,X4: real] :
      ( ( size_size_list_real @ ( append_real @ Xs2 @ ( cons_real @ X4 @ nil_real ) ) )
      = ( suc @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_8434_length__append__singleton,axiom,
    ! [Xs2: list_o,X4: $o] :
      ( ( size_size_list_o @ ( append_o @ Xs2 @ ( cons_o @ X4 @ nil_o ) ) )
      = ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_8435_length__append__singleton,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_8436_length__append__singleton,axiom,
    ! [Xs2: list_int,X4: int] :
      ( ( size_size_list_int @ ( append_int @ Xs2 @ ( cons_int @ X4 @ nil_int ) ) )
      = ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_8437_length__compl__rev__induct,axiom,
    ! [P: list_real > $o,L: list_real] :
      ( ( P @ nil_real )
     => ( ! [L2: list_real,E: real] :
            ( ! [Ll: list_real] :
                ( ( ord_less_eq_nat @ ( size_size_list_real @ Ll ) @ ( size_size_list_real @ L2 ) )
               => ( P @ Ll ) )
           => ( P @ ( append_real @ L2 @ ( cons_real @ E @ nil_real ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_8438_length__compl__rev__induct,axiom,
    ! [P: list_o > $o,L: list_o] :
      ( ( P @ nil_o )
     => ( ! [L2: list_o,E: $o] :
            ( ! [Ll: list_o] :
                ( ( ord_less_eq_nat @ ( size_size_list_o @ Ll ) @ ( size_size_list_o @ L2 ) )
               => ( P @ Ll ) )
           => ( P @ ( append_o @ L2 @ ( cons_o @ E @ nil_o ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_8439_length__compl__rev__induct,axiom,
    ! [P: list_nat > $o,L: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [L2: list_nat,E: nat] :
            ( ! [Ll: list_nat] :
                ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ll ) @ ( size_size_list_nat @ L2 ) )
               => ( P @ Ll ) )
           => ( P @ ( append_nat @ L2 @ ( cons_nat @ E @ nil_nat ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_8440_length__compl__rev__induct,axiom,
    ! [P: list_int > $o,L: list_int] :
      ( ( P @ nil_int )
     => ( ! [L2: list_int,E: int] :
            ( ! [Ll: list_int] :
                ( ( ord_less_eq_nat @ ( size_size_list_int @ Ll ) @ ( size_size_list_int @ L2 ) )
               => ( P @ Ll ) )
           => ( P @ ( append_int @ L2 @ ( cons_int @ E @ nil_int ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_8441_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X4: nat] :
      ( ( replicate_nat @ ( suc @ N ) @ X4 )
      = ( append_nat @ ( replicate_nat @ N @ X4 ) @ ( cons_nat @ X4 @ nil_nat ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_8442_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X4: int] :
      ( ( replicate_int @ ( suc @ N ) @ X4 )
      = ( append_int @ ( replicate_int @ N @ X4 ) @ ( cons_int @ X4 @ nil_int ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_8443_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X4: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT @ ( suc @ N ) @ X4 )
      = ( append_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X4 ) @ ( cons_VEBT_VEBT @ X4 @ nil_VEBT_VEBT ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_8444_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X4: $o] :
      ( ( replicate_o @ ( suc @ N ) @ X4 )
      = ( append_o @ ( replicate_o @ N @ X4 ) @ ( cons_o @ X4 @ nil_o ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_8445_comm__append__is__replicate,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( Xs2 != nil_o )
     => ( ( Ys != nil_o )
       => ( ( ( append_o @ Xs2 @ Ys )
            = ( append_o @ Ys @ Xs2 ) )
         => ? [N2: nat,Zs2: list_o] :
              ( ( ord_less_nat @ one_one_nat @ N2 )
              & ( ( concat_o @ ( replicate_list_o @ N2 @ Zs2 ) )
                = ( append_o @ Xs2 @ Ys ) ) ) ) ) ) ).

% comm_append_is_replicate
thf(fact_8446_comm__append__is__replicate,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( Ys != nil_nat )
       => ( ( ( append_nat @ Xs2 @ Ys )
            = ( append_nat @ Ys @ Xs2 ) )
         => ? [N2: nat,Zs2: list_nat] :
              ( ( ord_less_nat @ one_one_nat @ N2 )
              & ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
                = ( append_nat @ Xs2 @ Ys ) ) ) ) ) ) ).

% comm_append_is_replicate
thf(fact_8447_comm__append__is__replicate,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( Xs2 != nil_int )
     => ( ( Ys != nil_int )
       => ( ( ( append_int @ Xs2 @ Ys )
            = ( append_int @ Ys @ Xs2 ) )
         => ? [N2: nat,Zs2: list_int] :
              ( ( ord_less_nat @ one_one_nat @ N2 )
              & ( ( concat_int @ ( replicate_list_int @ N2 @ Zs2 ) )
                = ( append_int @ Xs2 @ Ys ) ) ) ) ) ) ).

% comm_append_is_replicate
thf(fact_8448_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_real,Ys: list_real] :
      ( ( enumerate_real @ N @ ( append_real @ Xs2 @ Ys ) )
      = ( append6678681742291297912t_real @ ( enumerate_real @ N @ Xs2 ) @ ( enumerate_real @ ( plus_plus_nat @ N @ ( size_size_list_real @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_8449_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_o] :
      ( ( enumerate_o @ N @ ( append_o @ Xs2 @ Ys ) )
      = ( append1535412585758129762_nat_o @ ( enumerate_o @ N @ Xs2 ) @ ( enumerate_o @ ( plus_plus_nat @ N @ ( size_size_list_o @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_8450_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs2 @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs2 ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_8451_enumerate__append__eq,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_int] :
      ( ( enumerate_int @ N @ ( append_int @ Xs2 @ Ys ) )
      = ( append6031344391939132024at_int @ ( enumerate_int @ N @ Xs2 ) @ ( enumerate_int @ ( plus_plus_nat @ N @ ( size_size_list_int @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_8452_n__lists_Osimps_I1_J,axiom,
    ! [Xs2: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs2 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_8453_n__lists_Osimps_I1_J,axiom,
    ! [Xs2: list_int] :
      ( ( n_lists_int @ zero_zero_nat @ Xs2 )
      = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% n_lists.simps(1)
thf(fact_8454_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8455_sum__nonpos,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8456_sum__nonpos,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8457_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8458_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8459_sum__nonpos,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8460_sum__nonpos,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8461_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8462_sum__nonpos,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8463_sum__nonpos,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8464_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8465_sum__nonneg,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8466_sum__nonneg,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8467_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8468_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8469_sum__nonneg,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8470_sum__nonneg,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8471_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8472_sum__nonneg,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8473_sum__nonneg,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8474_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_8475_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_8476_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_8477_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_8478_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_8479_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_8480_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_8481_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_8482_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_8483_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_8484_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A3 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A3 )
          = ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8485_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A3 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A3 )
          = ( groups6591440286371151544t_real @ G @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8486_sorted__append__bigger,axiom,
    ! [Xs2: list_real,Y: real] :
      ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( ord_less_eq_real @ X3 @ Y ) )
       => ( sorted_wrt_real @ ord_less_eq_real @ ( append_real @ Xs2 @ ( cons_real @ Y @ nil_real ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8487_sorted__append__bigger,axiom,
    ! [Xs2: list_o,Y: $o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs2 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( ord_less_eq_o @ X3 @ Y ) )
       => ( sorted_wrt_o @ ord_less_eq_o @ ( append_o @ Xs2 @ ( cons_o @ Y @ nil_o ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8488_sorted__append__bigger,axiom,
    ! [Xs2: list_rat,Y: rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs2 )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ ( set_rat2 @ Xs2 ) )
           => ( ord_less_eq_rat @ X3 @ Y ) )
       => ( sorted_wrt_rat @ ord_less_eq_rat @ ( append_rat @ Xs2 @ ( cons_rat @ Y @ nil_rat ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8489_sorted__append__bigger,axiom,
    ! [Xs2: list_num,Y: num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs2 )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ ( set_num2 @ Xs2 ) )
           => ( ord_less_eq_num @ X3 @ Y ) )
       => ( sorted_wrt_num @ ord_less_eq_num @ ( append_num @ Xs2 @ ( cons_num @ Y @ nil_num ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8490_sorted__append__bigger,axiom,
    ! [Xs2: list_nat,Y: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( ord_less_eq_nat @ X3 @ Y ) )
       => ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs2 @ ( cons_nat @ Y @ nil_nat ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8491_sorted__append__bigger,axiom,
    ! [Xs2: list_int,Y: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( ord_less_eq_int @ X3 @ Y ) )
       => ( sorted_wrt_int @ ord_less_eq_int @ ( append_int @ Xs2 @ ( cons_int @ Y @ nil_int ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_8492_zip__append1,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_nat] :
      ( ( zip_real_nat @ ( append_real @ Xs2 @ Ys ) @ Zs )
      = ( append2703634050814417656al_nat @ ( zip_real_nat @ Xs2 @ ( take_nat @ ( size_size_list_real @ Xs2 ) @ Zs ) ) @ ( zip_real_nat @ Ys @ ( drop_nat @ ( size_size_list_real @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8493_zip__append1,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_o] :
      ( ( zip_real_o @ ( append_real @ Xs2 @ Ys ) @ Zs )
      = ( append2727653632507881094real_o @ ( zip_real_o @ Xs2 @ ( take_o @ ( size_size_list_real @ Xs2 ) @ Zs ) ) @ ( zip_real_o @ Ys @ ( drop_o @ ( size_size_list_real @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8494_zip__append1,axiom,
    ! [Xs2: list_o,Ys: list_o,Zs: list_nat] :
      ( ( zip_o_nat @ ( append_o @ Xs2 @ Ys ) @ Zs )
      = ( append7250250098557412540_o_nat @ ( zip_o_nat @ Xs2 @ ( take_nat @ ( size_size_list_o @ Xs2 ) @ Zs ) ) @ ( zip_o_nat @ Ys @ ( drop_nat @ ( size_size_list_o @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8495_zip__append1,axiom,
    ! [Xs2: list_o,Ys: list_o,Zs: list_o] :
      ( ( zip_o_o @ ( append_o @ Xs2 @ Ys ) @ Zs )
      = ( append2614242729457001410od_o_o @ ( zip_o_o @ Xs2 @ ( take_o @ ( size_size_list_o @ Xs2 ) @ Zs ) ) @ ( zip_o_o @ Ys @ ( drop_o @ ( size_size_list_o @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8496_zip__append1,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) @ ( zip_nat_nat @ Ys @ ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8497_zip__append1,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_o] :
      ( ( zip_nat_o @ ( append_nat @ Xs2 @ Ys ) @ Zs )
      = ( append1535412585758129762_nat_o @ ( zip_nat_o @ Xs2 @ ( take_o @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) @ ( zip_nat_o @ Ys @ ( drop_o @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8498_zip__append1,axiom,
    ! [Xs2: list_int,Ys: list_int,Zs: list_nat] :
      ( ( zip_int_nat @ ( append_int @ Xs2 @ Ys ) @ Zs )
      = ( append1985177086494607480nt_nat @ ( zip_int_nat @ Xs2 @ ( take_nat @ ( size_size_list_int @ Xs2 ) @ Zs ) ) @ ( zip_int_nat @ Ys @ ( drop_nat @ ( size_size_list_int @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8499_zip__append1,axiom,
    ! [Xs2: list_int,Ys: list_int,Zs: list_o] :
      ( ( zip_int_o @ ( append_int @ Xs2 @ Ys ) @ Zs )
      = ( append8937742277180217606_int_o @ ( zip_int_o @ Xs2 @ ( take_o @ ( size_size_list_int @ Xs2 ) @ Zs ) ) @ ( zip_int_o @ Ys @ ( drop_o @ ( size_size_list_int @ Xs2 ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_8500_zip__append2,axiom,
    ! [Xs2: list_nat,Ys: list_real,Zs: list_real] :
      ( ( zip_nat_real @ Xs2 @ ( append_real @ Ys @ Zs ) )
      = ( append6678681742291297912t_real @ ( zip_nat_real @ ( take_nat @ ( size_size_list_real @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_nat_real @ ( drop_nat @ ( size_size_list_real @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8501_zip__append2,axiom,
    ! [Xs2: list_o,Ys: list_real,Zs: list_real] :
      ( ( zip_o_real @ Xs2 @ ( append_real @ Ys @ Zs ) )
      = ( append131710322270406936o_real @ ( zip_o_real @ ( take_o @ ( size_size_list_real @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_o_real @ ( drop_o @ ( size_size_list_real @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8502_zip__append2,axiom,
    ! [Xs2: list_nat,Ys: list_o,Zs: list_o] :
      ( ( zip_nat_o @ Xs2 @ ( append_o @ Ys @ Zs ) )
      = ( append1535412585758129762_nat_o @ ( zip_nat_o @ ( take_nat @ ( size_size_list_o @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_nat_o @ ( drop_nat @ ( size_size_list_o @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8503_zip__append2,axiom,
    ! [Xs2: list_o,Ys: list_o,Zs: list_o] :
      ( ( zip_o_o @ Xs2 @ ( append_o @ Ys @ Zs ) )
      = ( append2614242729457001410od_o_o @ ( zip_o_o @ ( take_o @ ( size_size_list_o @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_o_o @ ( drop_o @ ( size_size_list_o @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8504_zip__append2,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_nat_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) )
      = ( append985823374593552924at_nat @ ( zip_nat_nat @ ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_nat_nat @ ( drop_nat @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8505_zip__append2,axiom,
    ! [Xs2: list_o,Ys: list_nat,Zs: list_nat] :
      ( ( zip_o_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) )
      = ( append7250250098557412540_o_nat @ ( zip_o_nat @ ( take_o @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_o_nat @ ( drop_o @ ( size_size_list_nat @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8506_zip__append2,axiom,
    ! [Xs2: list_nat,Ys: list_int,Zs: list_int] :
      ( ( zip_nat_int @ Xs2 @ ( append_int @ Ys @ Zs ) )
      = ( append6031344391939132024at_int @ ( zip_nat_int @ ( take_nat @ ( size_size_list_int @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_nat_int @ ( drop_nat @ ( size_size_list_int @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8507_zip__append2,axiom,
    ! [Xs2: list_o,Ys: list_int,Zs: list_int] :
      ( ( zip_o_int @ Xs2 @ ( append_int @ Ys @ Zs ) )
      = ( append3072399079048215832_o_int @ ( zip_o_int @ ( take_o @ ( size_size_list_int @ Ys ) @ Xs2 ) @ Ys ) @ ( zip_o_int @ ( drop_o @ ( size_size_list_int @ Ys ) @ Xs2 ) @ Zs ) ) ) ).

% zip_append2
thf(fact_8508_list__rest__coinc,axiom,
    ! [S22: list_real,S1: list_real,R12: list_real,R23: list_real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ S22 ) @ ( size_size_list_real @ S1 ) )
     => ( ( ( append_real @ S1 @ R12 )
          = ( append_real @ S22 @ R23 ) )
       => ? [R1p: list_real] :
            ( R23
            = ( append_real @ R1p @ R12 ) ) ) ) ).

% list_rest_coinc
thf(fact_8509_list__rest__coinc,axiom,
    ! [S22: list_o,S1: list_o,R12: list_o,R23: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ S22 ) @ ( size_size_list_o @ S1 ) )
     => ( ( ( append_o @ S1 @ R12 )
          = ( append_o @ S22 @ R23 ) )
       => ? [R1p: list_o] :
            ( R23
            = ( append_o @ R1p @ R12 ) ) ) ) ).

% list_rest_coinc
thf(fact_8510_list__rest__coinc,axiom,
    ! [S22: list_nat,S1: list_nat,R12: list_nat,R23: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ S22 ) @ ( size_size_list_nat @ S1 ) )
     => ( ( ( append_nat @ S1 @ R12 )
          = ( append_nat @ S22 @ R23 ) )
       => ? [R1p: list_nat] :
            ( R23
            = ( append_nat @ R1p @ R12 ) ) ) ) ).

% list_rest_coinc
thf(fact_8511_list__rest__coinc,axiom,
    ! [S22: list_int,S1: list_int,R12: list_int,R23: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ S22 ) @ ( size_size_list_int @ S1 ) )
     => ( ( ( append_int @ S1 @ R12 )
          = ( append_int @ S22 @ R23 ) )
       => ? [R1p: list_int] :
            ( R23
            = ( append_int @ R1p @ R12 ) ) ) ) ).

% list_rest_coinc
thf(fact_8512_sum_Ointer__filter,axiom,
    ! [A3: set_VEBT_VEBT,G: vEBT_VEBT > complex,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( groups1794756597179926696omplex @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups1794756597179926696omplex
          @ ^ [X: vEBT_VEBT] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8513_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X: real] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8514_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X: nat] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8515_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X: int] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8516_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > complex,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups8024822376189712711omplex @ G
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8024822376189712711omplex
          @ ^ [X: code_integer] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8517_sum_Ointer__filter,axiom,
    ! [A3: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( groups2240296850493347238T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups2240296850493347238T_real
          @ ^ [X: vEBT_VEBT] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8518_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8519_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8520_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8521_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > real,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups1270011288395367621r_real
          @ ^ [X: code_integer] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8522_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_real @ nil_real )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_8523_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_o @ nil_o )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_8524_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_8525_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_int @ nil_int )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_8526_len__greater__imp__nonempty,axiom,
    ! [X4: nat,L: list_real] :
      ( ( ord_less_nat @ X4 @ ( size_size_list_real @ L ) )
     => ( L != nil_real ) ) ).

% len_greater_imp_nonempty
thf(fact_8527_len__greater__imp__nonempty,axiom,
    ! [X4: nat,L: list_o] :
      ( ( ord_less_nat @ X4 @ ( size_size_list_o @ L ) )
     => ( L != nil_o ) ) ).

% len_greater_imp_nonempty
thf(fact_8528_len__greater__imp__nonempty,axiom,
    ! [X4: nat,L: list_nat] :
      ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ L ) )
     => ( L != nil_nat ) ) ).

% len_greater_imp_nonempty
thf(fact_8529_len__greater__imp__nonempty,axiom,
    ! [X4: nat,L: list_int] :
      ( ( ord_less_nat @ X4 @ ( size_size_list_int @ L ) )
     => ( L != nil_int ) ) ).

% len_greater_imp_nonempty
thf(fact_8530_list__induct2,axiom,
    ! [Xs2: list_real,Ys: list_real,P: list_real > list_real > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( P @ nil_real @ nil_real )
       => ( ! [X3: real,Xs3: list_real,Y3: real,Ys5: list_real] :
              ( ( ( size_size_list_real @ Xs3 )
                = ( size_size_list_real @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8531_list__induct2,axiom,
    ! [Xs2: list_real,Ys: list_o,P: list_real > list_o > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( P @ nil_real @ nil_o )
       => ( ! [X3: real,Xs3: list_real,Y3: $o,Ys5: list_o] :
              ( ( ( size_size_list_real @ Xs3 )
                = ( size_size_list_o @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8532_list__induct2,axiom,
    ! [Xs2: list_real,Ys: list_nat,P: list_real > list_nat > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_real @ nil_nat )
       => ( ! [X3: real,Xs3: list_real,Y3: nat,Ys5: list_nat] :
              ( ( ( size_size_list_real @ Xs3 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8533_list__induct2,axiom,
    ! [Xs2: list_real,Ys: list_int,P: list_real > list_int > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_real @ nil_int )
       => ( ! [X3: real,Xs3: list_real,Y3: int,Ys5: list_int] :
              ( ( ( size_size_list_real @ Xs3 )
                = ( size_size_list_int @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8534_list__induct2,axiom,
    ! [Xs2: list_o,Ys: list_real,P: list_o > list_real > $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( P @ nil_o @ nil_real )
       => ( ! [X3: $o,Xs3: list_o,Y3: real,Ys5: list_real] :
              ( ( ( size_size_list_o @ Xs3 )
                = ( size_size_list_real @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8535_list__induct2,axiom,
    ! [Xs2: list_o,Ys: list_o,P: list_o > list_o > $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( P @ nil_o @ nil_o )
       => ( ! [X3: $o,Xs3: list_o,Y3: $o,Ys5: list_o] :
              ( ( ( size_size_list_o @ Xs3 )
                = ( size_size_list_o @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8536_list__induct2,axiom,
    ! [Xs2: list_o,Ys: list_nat,P: list_o > list_nat > $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_o @ nil_nat )
       => ( ! [X3: $o,Xs3: list_o,Y3: nat,Ys5: list_nat] :
              ( ( ( size_size_list_o @ Xs3 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8537_list__induct2,axiom,
    ! [Xs2: list_o,Ys: list_int,P: list_o > list_int > $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_o @ nil_int )
       => ( ! [X3: $o,Xs3: list_o,Y3: int,Ys5: list_int] :
              ( ( ( size_size_list_o @ Xs3 )
                = ( size_size_list_int @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8538_list__induct2,axiom,
    ! [Xs2: list_nat,Ys: list_real,P: list_nat > list_real > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( P @ nil_nat @ nil_real )
       => ( ! [X3: nat,Xs3: list_nat,Y3: real,Ys5: list_real] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_real @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8539_list__induct2,axiom,
    ! [Xs2: list_nat,Ys: list_o,P: list_nat > list_o > $o] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( P @ nil_nat @ nil_o )
       => ( ! [X3: nat,Xs3: list_nat,Y3: $o,Ys5: list_o] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_o @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_8540_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_real,P: list_real > list_real > list_real > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( P @ nil_real @ nil_real @ nil_real )
         => ( ! [X3: real,Xs3: list_real,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_real @ Ys5 ) )
               => ( ( ( size_size_list_real @ Ys5 )
                    = ( size_size_list_real @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8541_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_o,P: list_real > list_real > list_o > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( P @ nil_real @ nil_real @ nil_o )
         => ( ! [X3: real,Xs3: list_real,Y3: real,Ys5: list_real,Z3: $o,Zs2: list_o] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_real @ Ys5 ) )
               => ( ( ( size_size_list_real @ Ys5 )
                    = ( size_size_list_o @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8542_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_nat,P: list_real > list_real > list_nat > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_real @ nil_real @ nil_nat )
         => ( ! [X3: real,Xs3: list_real,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_real @ Ys5 ) )
               => ( ( ( size_size_list_real @ Ys5 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8543_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_real,Zs: list_int,P: list_real > list_real > list_int > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_real @ nil_real @ nil_int )
         => ( ! [X3: real,Xs3: list_real,Y3: real,Ys5: list_real,Z3: int,Zs2: list_int] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_real @ Ys5 ) )
               => ( ( ( size_size_list_real @ Ys5 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8544_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_o,Zs: list_real,P: list_real > list_o > list_real > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( P @ nil_real @ nil_o @ nil_real )
         => ( ! [X3: real,Xs3: list_real,Y3: $o,Ys5: list_o,Z3: real,Zs2: list_real] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_o @ Ys5 ) )
               => ( ( ( size_size_list_o @ Ys5 )
                    = ( size_size_list_real @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8545_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_o,Zs: list_o,P: list_real > list_o > list_o > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( P @ nil_real @ nil_o @ nil_o )
         => ( ! [X3: real,Xs3: list_real,Y3: $o,Ys5: list_o,Z3: $o,Zs2: list_o] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_o @ Ys5 ) )
               => ( ( ( size_size_list_o @ Ys5 )
                    = ( size_size_list_o @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8546_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_o,Zs: list_nat,P: list_real > list_o > list_nat > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_real @ nil_o @ nil_nat )
         => ( ! [X3: real,Xs3: list_real,Y3: $o,Ys5: list_o,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_o @ Ys5 ) )
               => ( ( ( size_size_list_o @ Ys5 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8547_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_o,Zs: list_int,P: list_real > list_o > list_int > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_real @ nil_o @ nil_int )
         => ( ! [X3: real,Xs3: list_real,Y3: $o,Ys5: list_o,Z3: int,Zs2: list_int] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_o @ Ys5 ) )
               => ( ( ( size_size_list_o @ Ys5 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_o @ Y3 @ Ys5 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8548_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_nat,Zs: list_real,P: list_real > list_nat > list_real > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( P @ nil_real @ nil_nat @ nil_real )
         => ( ! [X3: real,Xs3: list_real,Y3: nat,Ys5: list_nat,Z3: real,Zs2: list_real] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_real @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8549_list__induct3,axiom,
    ! [Xs2: list_real,Ys: list_nat,Zs: list_o,P: list_real > list_nat > list_o > $o] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( P @ nil_real @ nil_nat @ nil_o )
         => ( ! [X3: real,Xs3: list_real,Y3: nat,Ys5: list_nat,Z3: $o,Zs2: list_o] :
                ( ( ( size_size_list_real @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_o @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_real @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_8550_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_real,Ws: list_real,P: list_int > list_real > list_real > list_real > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( ( size_size_list_real @ Zs )
            = ( size_size_list_real @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_real @ nil_real )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: real,Ws2: list_real] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_real @ Zs2 ) )
                   => ( ( ( size_size_list_real @ Zs2 )
                        = ( size_size_list_real @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8551_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_real,Ws: list_o,P: list_int > list_real > list_real > list_o > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( ( size_size_list_real @ Zs )
            = ( size_size_list_o @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_real @ nil_o )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: $o,Ws2: list_o] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_real @ Zs2 ) )
                   => ( ( ( size_size_list_real @ Zs2 )
                        = ( size_size_list_o @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8552_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_real,Ws: list_nat,P: list_int > list_real > list_real > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( ( size_size_list_real @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_real @ nil_nat )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_real @ Zs2 ) )
                   => ( ( ( size_size_list_real @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8553_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_real,Ws: list_int,P: list_int > list_real > list_real > list_int > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_real @ Zs ) )
       => ( ( ( size_size_list_real @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_real @ nil_int )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: real,Zs2: list_real,W: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_real @ Zs2 ) )
                   => ( ( ( size_size_list_real @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_real @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8554_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_o,Ws: list_real,P: list_int > list_real > list_o > list_real > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( ( size_size_list_o @ Zs )
            = ( size_size_list_real @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_o @ nil_real )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: $o,Zs2: list_o,W: real,Ws2: list_real] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_o @ Zs2 ) )
                   => ( ( ( size_size_list_o @ Zs2 )
                        = ( size_size_list_real @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8555_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_o,Ws: list_o,P: list_int > list_real > list_o > list_o > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( ( size_size_list_o @ Zs )
            = ( size_size_list_o @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_o @ nil_o )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: $o,Zs2: list_o,W: $o,Ws2: list_o] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_o @ Zs2 ) )
                   => ( ( ( size_size_list_o @ Zs2 )
                        = ( size_size_list_o @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8556_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_o,Ws: list_nat,P: list_int > list_real > list_o > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( ( size_size_list_o @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_o @ nil_nat )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: $o,Zs2: list_o,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_o @ Zs2 ) )
                   => ( ( ( size_size_list_o @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8557_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_o,Ws: list_int,P: list_int > list_real > list_o > list_int > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_o @ Zs ) )
       => ( ( ( size_size_list_o @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_o @ nil_int )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: $o,Zs2: list_o,W: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_o @ Zs2 ) )
                   => ( ( ( size_size_list_o @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8558_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_nat,Ws: list_real,P: list_int > list_real > list_nat > list_real > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_real @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_nat @ nil_real )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat,W: real,Ws2: list_real] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_real @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8559_list__induct4,axiom,
    ! [Xs2: list_int,Ys: list_real,Zs: list_nat,Ws: list_o,P: list_int > list_real > list_nat > list_o > $o] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( ( size_size_list_real @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_o @ Ws ) )
         => ( ( P @ nil_int @ nil_real @ nil_nat @ nil_o )
           => ( ! [X3: int,Xs3: list_int,Y3: real,Ys5: list_real,Z3: nat,Zs2: list_nat,W: $o,Ws2: list_o] :
                  ( ( ( size_size_list_int @ Xs3 )
                    = ( size_size_list_real @ Ys5 ) )
                 => ( ( ( size_size_list_real @ Ys5 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_o @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_int @ X3 @ Xs3 ) @ ( cons_real @ Y3 @ Ys5 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_8560_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_8561_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8562_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8563_ln__inj__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X4 )
            = ( ln_ln_real @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_8564_ln__less__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_8565_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_8566_ln__le__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_8567_ln__eq__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = zero_zero_real )
        = ( X4 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_8568_ln__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_iff
thf(fact_8569_ln__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_8570_ln__ge__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_iff
thf(fact_8571_ln__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_8572_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8573_ln__less__self,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_less_self
thf(fact_8574_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_8575_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S7: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_8576_ln__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_bound
thf(fact_8577_ln__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_gt_zero
thf(fact_8578_ln__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_8579_ln__gt__zero__imp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_8580_ln__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_ge_zero
thf(fact_8581_ln__ge__zero__imp__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_8582_ln__add__one__self__le__self,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self
thf(fact_8583_ln__mult,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X4 @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_8584_ln__eq__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = ( minus_minus_real @ X4 @ one_one_real ) )
       => ( X4 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_8585_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_8586_ln__le__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_8587_ln__add__one__self__le__self2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self2
thf(fact_8588_ln__realpow,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( power_power_real @ X4 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_realpow
thf(fact_8589_ln__one__minus__pos__upper__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) @ ( uminus_uminus_real @ X4 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_8590_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X4: 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 @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R2: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R2 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R2 ) )
                @ ( power_power_nat @ X4 @ R2 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8591_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_8592_list__encode_Ocases,axiom,
    ! [X4: list_nat] :
      ( ( X4 != nil_nat )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( X4
           != ( cons_nat @ X3 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_8593_real__of__nat__div4,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% real_of_nat_div4
thf(fact_8594_real__of__nat__div2,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) ) ).

% real_of_nat_div2
thf(fact_8595_real__of__nat__div3,axiom,
    ! [N: nat,X4: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X4 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_8596_complex__mod__minus__le__complex__mod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_8597_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_8598_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_8599_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_8600_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_8601_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_8602_zdiv__mono1,axiom,
    ! [A: int,A6: int,B: int] :
      ( ( ord_less_eq_int @ A @ A6 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_8603_zdiv__mono2,axiom,
    ! [A: int,B6: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B6 )
       => ( ( ord_less_eq_int @ B6 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_8604_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_8605_zdiv__mono1__neg,axiom,
    ! [A: int,A6: int,B: int] :
      ( ( ord_less_eq_int @ A @ A6 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_8606_zdiv__mono2__neg,axiom,
    ! [A: int,B6: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B6 )
       => ( ( ord_less_eq_int @ B6 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_8607_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_8608_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_8609_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_8610_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_8611_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_8612_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_8613_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_8614_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_8615_int__div__less__self,axiom,
    ! [X4: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).

% int_div_less_self
thf(fact_8616_ln__div,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X4 @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_8617_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_8618_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_8619_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S7: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_8620_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_eq_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B4 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_8621_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_8622_ln__diff__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X4 @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_8623_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_8624_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q5: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q5 ) ) ) ) ).

% int_div_pos_eq
thf(fact_8625_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q5: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( divide_divide_int @ A @ B )
            = Q5 ) ) ) ) ).

% int_div_neg_eq
thf(fact_8626_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [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 ) ) )
             => ( P @ 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 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_8627_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_8628_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_8629_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_8630_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_8631_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_8632_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_8633_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_8634_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_8635_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_8636_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_8637_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_8638_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_8639_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_8640_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q5 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q5 ) ) ).

% div_mult2_eq
thf(fact_8641_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_8642_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_8643_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_8644_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_8645_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_8646_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_8647_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_8648_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_8649_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_8650_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A @ C ) @ B ) ) ) ).

% td_gal_lt
thf(fact_8651_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q5 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q5 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q5 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_8652_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_8653_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_8654_zdiv__mult__self,axiom,
    ! [M: int,A: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ M @ N ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A @ M ) @ N ) ) ) ).

% zdiv_mult_self
thf(fact_8655_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N4 )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).

% div_if
thf(fact_8656_div__nat__eqI,axiom,
    ! [N: nat,Q5: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q5 ) ) ) ).

% div_nat_eqI
thf(fact_8657_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_8658_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_8659_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_8660_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B @ C ) @ A )
        = ( ord_less_eq_nat @ B @ ( divide_divide_nat @ A @ C ) ) ) ) ).

% td_gal
thf(fact_8661_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q5 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q5 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q5 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_8662_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_8663_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_8664_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q7: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q7 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q7 ) ) )
            & ( P @ Q7 ) ) ) ) ).

% split_div'
thf(fact_8665_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_8666_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_8667_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( divide_divide_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( divide_divide_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_8668_list__encode_Oelims,axiom,
    ! [X4: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y )
     => ( ( ( X4 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs3: list_nat] :
              ( ( X4
                = ( cons_nat @ X3 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_8669_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_8670_list__encode__eq,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( ( nat_list_encode @ X4 )
        = ( nat_list_encode @ Y ) )
      = ( X4 = Y ) ) ).

% list_encode_eq
thf(fact_8671_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_8672_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_8673_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_8674_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_8675_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_8676_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_8677_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_8678_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_8679_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_8680_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_8681_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_8682_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_8683_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_8684_list__encode_Osimps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X4 @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_8685_list__encode_Opelims,axiom,
    ! [X4: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X4 )
       => ( ( ( X4 = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( X4
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_8686_arctan__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X4 ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X4 @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_8687_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_8688_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_8689_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_8690_arctan__eq__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( arctan @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_8691_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_8692_arctan__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_8693_zero__less__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% zero_less_arctan_iff
thf(fact_8694_zero__le__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_arctan_iff
thf(fact_8695_arctan__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_8696_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8697_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_8698_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_8699_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_8700_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_8701_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_8702_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_8703_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_8704_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_8705_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_8706_arctan__monotone_H,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_8707_arctan__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% arctan_le_iff
thf(fact_8708_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_8709_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P5: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P5 )
       => ( ( ord_less_nat @ M @ P5 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P5 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_8710_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_8711_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_8712_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
     => ( ! [M2: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M2 @ N2 ) )
             => ( P @ M2 @ N2 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_8713_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_8714_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_8715_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_8716_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_8717_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8718_mod__nat__sub,axiom,
    ! [X4: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X4 @ Y ) @ Z )
        = ( minus_minus_nat @ X4 @ Y ) ) ) ).

% mod_nat_sub
thf(fact_8719_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N4 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).

% mod_if
thf(fact_8720_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_8721_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q4: nat] :
          ( M
          = ( times_times_nat @ D @ Q4 ) ) ) ).

% mod_eq_0D
thf(fact_8722_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_8723_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8724_nat__mod__eq__iff,axiom,
    ! [X4: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q12: nat,Q23: nat] :
            ( ( plus_plus_nat @ X4 @ ( times_times_nat @ N @ Q12 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q23 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_8725_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% msrevs(2)
thf(fact_8726_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8727_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S7: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8728_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_8729_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_8730_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_8731_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_8732_div__less__mono,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A3 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B4 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_8733_mod__nat__add,axiom,
    ! [X4: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Z )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z )
              = ( plus_plus_nat @ X4 @ Y ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_8734_nat__mod__eq__lemma,axiom,
    ! [X4: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X4 )
       => ? [Q4: nat] :
            ( X4
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_8735_mod__eq__nat2E,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q5 )
        = ( modulo_modulo_nat @ N @ Q5 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q5 @ S ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_8736_mod__eq__nat1E,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q5 )
        = ( modulo_modulo_nat @ N @ Q5 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q5 @ S ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_8737_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q5 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q5 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_8738_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] :
      ( A3
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N ) @ N ) @ ( modulo_modulo_nat @ A3 @ N ) ) ) ).

% div_mod_decomp
thf(fact_8739_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N4: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).

% modulo_nat_def
thf(fact_8740_mod__lemma,axiom,
    ! [C: nat,R3: nat,B: nat,Q5: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R3 @ B )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ Q5 @ C ) ) @ R3 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mod_lemma
thf(fact_8741_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_8742_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_8743_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_8744_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X6
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X6 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_8745_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X6
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X6 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_8746_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X6
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( plus_plus_int @ X6 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_8747_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X6
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( plus_plus_int @ X6 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_8748_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X6
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X6 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_8749_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X6
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X6 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_8750_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X6
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( minus_minus_int @ X6 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_8751_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 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X6
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( minus_minus_int @ X6 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_8752_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_8753_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_8754_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X6
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X6 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_8755_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X6
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X6 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_8756_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 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B4 )
                   => ( X6
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X6 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_8757_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B4 )
                 => ( X6
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X6 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_8758_cppi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P4 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A3 )
                     => ( X3
                       != ( minus_minus_int @ Xb3 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P4 @ X3 )
                = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ A3 )
                        & ( P @ ( minus_minus_int @ Y4 @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_8759_cpmi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P4 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B4 )
                     => ( X3
                       != ( plus_plus_int @ Xb3 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P4 @ X3 )
                = ( P4 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ B4 )
                        & ( P @ ( plus_plus_int @ Y4 @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_8760_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ 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_8761_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_8762_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_8763_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_eq_real @ X4 @ ( 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 @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8764_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X4 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8765_exp__le__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_8766_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_8767_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_8768_exp__eq__one__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ X4 )
        = one_one_real )
      = ( X4 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_8769_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8770_one__less__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% one_less_exp_iff
thf(fact_8771_exp__less__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_8772_exp__le__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_8773_one__le__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% one_le_exp_iff
thf(fact_8774_exp__ln__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% exp_ln_iff
thf(fact_8775_exp__ln,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 ) ) ).

% exp_ln
thf(fact_8776_Word_Omod__minus__cong,axiom,
    ! [B: int,B6: int,X4: int,X9: int,Y: int,Y8: int,Z7: int] :
      ( ( B = B6 )
     => ( ( ( modulo_modulo_int @ X4 @ B6 )
          = ( modulo_modulo_int @ X9 @ B6 ) )
       => ( ( ( modulo_modulo_int @ Y @ B6 )
            = ( modulo_modulo_int @ Y8 @ B6 ) )
         => ( ( ( minus_minus_int @ X9 @ Y8 )
              = Z7 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X4 @ Y ) @ B )
              = ( modulo_modulo_int @ Z7 @ B6 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_8777_mod__plus__cong,axiom,
    ! [B: int,B6: int,X4: int,X9: int,Y: int,Y8: int,Z7: int] :
      ( ( B = B6 )
     => ( ( ( modulo_modulo_int @ X4 @ B6 )
          = ( modulo_modulo_int @ X9 @ B6 ) )
       => ( ( ( modulo_modulo_int @ Y @ B6 )
            = ( modulo_modulo_int @ Y8 @ B6 ) )
         => ( ( ( plus_plus_int @ X9 @ Y8 )
              = Z7 )
           => ( ( modulo_modulo_int @ ( plus_plus_int @ X4 @ Y ) @ B )
              = ( modulo_modulo_int @ Z7 @ B6 ) ) ) ) ) ) ).

% mod_plus_cong
thf(fact_8778_not__exp__less__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_8779_exp__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_gt_zero
thf(fact_8780_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y ) ) ).

% exp_total
thf(fact_8781_exp__ge__zero,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_ge_zero
thf(fact_8782_not__exp__le__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_8783_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_8784_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_8785_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_8786_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_8787_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_8788_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q4: int] :
          ( M
          = ( times_times_int @ D @ Q4 ) ) ) ).

% zmod_eq_0D
thf(fact_8789_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q7: int] :
            ( M
            = ( times_times_int @ D @ Q7 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_8790_exp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) ) ) ).

% exp_gt_one
thf(fact_8791_exp__ge__add__one__self,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ).

% exp_ge_add_one_self
thf(fact_8792_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_8793_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_8794_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_8795_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_8796_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_8797_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_8798_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_8799_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_8800_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_8801_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_8802_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_8803_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_8804_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A3 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B4 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B4 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_8805_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_8806_exp__ge__add__one__self__aux,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_8807_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_8808_ln__ge__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X4 ) ) ) ).

% ln_ge_iff
thf(fact_8809_ln__x__over__x__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_8810_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_8811_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_8812_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_8813_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_8814_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q5: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_8815_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q5: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_8816_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [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 ) ) )
             => ( P @ 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 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_8817_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_8818_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_8819_mod__add__if__z,axiom,
    ! [X4: int,Z: int,Y: int] :
      ( ( ord_less_int @ X4 @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X4 @ Y ) @ Z )
                    = ( plus_plus_int @ X4 @ Y ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X4 @ Y ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X4 @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_8820_mod__sub__if__z,axiom,
    ! [X4: int,Z: int,Y: int] :
      ( ( ord_less_int @ X4 @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y @ X4 )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X4 @ Y ) @ Z )
                    = ( minus_minus_int @ X4 @ Y ) ) )
                & ( ~ ( ord_less_eq_int @ Y @ X4 )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X4 @ Y ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X4 @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_8821_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_8822_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_8823_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_8824_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S7: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_8825_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,N: int] :
      ( ( ord_less_int @ A3 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ 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_8826_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [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 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_8827_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [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 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_8828_Maclaurin__exp__lt,axiom,
    ! [X4: real,N: nat] :
      ( ( X4 != 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 @ X4 ) )
            & ( ( exp_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8829_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_8830_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_8831_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_8832_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_8833_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_8834_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_8835_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_8836_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_8837_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_8838_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_8839_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_8840_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q5: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q5 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q5 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_8841_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_8842_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_8843_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_8844_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_8845_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs2 @ Ys ) )
        = ( ? [K3: nat] :
              ( ( ord_less_eq_nat @ I @ K3 )
              & ( ord_less_eq_nat @ K3 @ J )
              & ( ( upt @ I @ K3 )
                = Xs2 )
              & ( ( upt @ K3 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_8846_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_8847_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_8848_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_8849_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M5 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_8850_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_8851_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).

% atLeast_upt
thf(fact_8852_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X4: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X4 @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X4 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_8853_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).

% atMost_upto
thf(fact_8854_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_8855_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_8856_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_8857_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_8858_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B9: 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 @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8859_Maclaurin__exp__le,axiom,
    ! [X4: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X4 ) )
      & ( ( exp_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8860_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_8861_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_8862_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_8863_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_8864_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_8865_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_8866_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R3 ) ) ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8867_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I2: nat] :
            ( ( ( ord_less_eq_nat @ U @ I2 )
              & ( ord_less_nat @ I2 @ U3 ) )
           => ~ ( P @ I2 ) )
       => ( ( filter_nat @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_8868_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_8869_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_8870_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8871_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_8872_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_8873_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8874_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_8875_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8876_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_8877_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_8878_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_8879_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R3 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_8880_binomial__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R3 ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_8881_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_8882_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_8883_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_8884_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_8885_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_8886_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_8887_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_8888_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_8889_sum__choose__lower,axiom,
    ! [R3: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_8890_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_8891_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_8892_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_8893_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_8894_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_8895_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_8896_vandermonde,axiom,
    ! [M: nat,N: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R3 ) ) ).

% vandermonde
thf(fact_8897_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_8898_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_8899_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] : N4 ) ) ).

% of_nat_id
thf(fact_8900_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_8901_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_8902_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8903_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_8904_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_8905_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_8906_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( inf_inf_assn @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_ctx_iff
thf(fact_8907_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( inf_inf_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_ctx_iff
thf(fact_8908_smod__int__0__mod,axiom,
    ! [X4: int] :
      ( ( signed6292675348222524329lo_int @ zero_zero_int @ X4 )
      = zero_zero_int ) ).

% smod_int_0_mod
thf(fact_8909_smod__int__mod__0,axiom,
    ! [X4: int] :
      ( ( signed6292675348222524329lo_int @ X4 @ zero_zero_int )
      = X4 ) ).

% smod_int_mod_0
thf(fact_8910_and__extract__pure__right__iff,axiom,
    ! [P: assn,B: $o] :
      ( ( inf_inf_assn @ P @ ( pure_assn @ B ) )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ P ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_iff
thf(fact_8911_and__extract__pure__left__iff,axiom,
    ! [B: $o,Q: assn] :
      ( ( inf_inf_assn @ ( pure_assn @ B ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_iff
thf(fact_8912_ent__conjE2,axiom,
    ! [B4: assn,C2: assn,A3: assn] :
      ( ( entails @ B4 @ C2 )
     => ( entails @ ( inf_inf_assn @ A3 @ B4 ) @ C2 ) ) ).

% ent_conjE2
thf(fact_8913_ent__conjE1,axiom,
    ! [A3: assn,C2: assn,B4: assn] :
      ( ( entails @ A3 @ C2 )
     => ( entails @ ( inf_inf_assn @ A3 @ B4 ) @ C2 ) ) ).

% ent_conjE1
thf(fact_8914_ent__conjI,axiom,
    ! [A3: assn,B4: assn,C2: assn] :
      ( ( entails @ A3 @ B4 )
     => ( ( entails @ A3 @ C2 )
       => ( entails @ A3 @ ( inf_inf_assn @ B4 @ C2 ) ) ) ) ).

% ent_conjI
thf(fact_8915_norm__assertion__simps_I9_J,axiom,
    ! [X4: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X4 )
      = bot_bot_assn ) ).

% norm_assertion_simps(9)
thf(fact_8916_norm__assertion__simps_I10_J,axiom,
    ! [X4: assn] :
      ( ( inf_inf_assn @ X4 @ bot_bot_assn )
      = bot_bot_assn ) ).

% norm_assertion_simps(10)
thf(fact_8917_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_8918_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_8919_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_8920_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_8921_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_8922_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_8923_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_8924_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_8925_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_8926_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_8927_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_8928_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
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8929_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
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8930_upto_Opelims,axiom,
    ! [X4: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
               => ( Y
                  = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X4 @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_8931_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_8932_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_8933_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( A2
          = ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% less_eq_assn_def
thf(fact_8934_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A2: assn,B2: assn] : ( inf_inf_assn @ A2 @ ( uminus_uminus_assn @ B2 ) ) ) ) ).

% minus_assn_def
thf(fact_8935_assn__aci_I4_J,axiom,
    ! [X4: assn,Y: assn] :
      ( ( inf_inf_assn @ X4 @ ( inf_inf_assn @ X4 @ Y ) )
      = ( inf_inf_assn @ X4 @ Y ) ) ).

% assn_aci(4)
thf(fact_8936_assn__aci_I3_J,axiom,
    ! [X4: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X4 @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X4 @ Z ) ) ) ).

% assn_aci(3)
thf(fact_8937_assn__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X: assn,Y4: assn] : ( inf_inf_assn @ Y4 @ X ) ) ) ).

% assn_aci(1)
thf(fact_8938_norm__assertion__simps_I31_J,axiom,
    ! [X4: assn] :
      ( ( inf_inf_assn @ X4 @ X4 )
      = X4 ) ).

% norm_assertion_simps(31)
thf(fact_8939_norm__assertion__simps_I14_J,axiom,
    ! [X4: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X4 @ Y ) @ Z )
      = ( inf_inf_assn @ X4 @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(14)
thf(fact_8940_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_8941_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_8942_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_8943_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_8944_upto_Oelims,axiom,
    ! [X4: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X4 @ Xa )
         => ( Y
            = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X4 @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_8945_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_8946_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_8947_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_8948_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8949_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_8950_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_8951_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_8952_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_8953_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_8954_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% less_assn_def
thf(fact_8955_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_8956_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_8957_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_8958_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_8959_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_8960_real__of__int__div4,axiom,
    ! [N: int,X4: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) ) ).

% real_of_int_div4
thf(fact_8961_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N4: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_8962_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N4: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_8963_real__of__int__div2,axiom,
    ! [N: int,X4: 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 @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) ) ).

% real_of_int_div2
thf(fact_8964_real__of__int__div3,axiom,
    ! [N: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X4 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_8965_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_8966_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_8967_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 )
          & ? [M2: nat] :
              ( ( ( some_nat @ M2 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_8968_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_8969_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_8970_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_8971_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_8972_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X4: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X4 )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_8973_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_8974_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_8975_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_8976_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_8977_insert__simp__mima,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X4 = Mi )
        | ( X4 = 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 ) @ X4 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_8978_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X4 )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_8979_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X4 )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_8980_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( 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 @ X4 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X4 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_8981_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X4 ) @ X4 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_8982_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X4 @ ( 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 @ X4 ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X4 = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_8983_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_8984_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_8985_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_8986_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_8987_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_8988_delt__out__of__range,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X4 @ Mi )
        | ( ord_less_nat @ Ma @ X4 ) )
     => ( ( 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 ) @ X4 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_8989_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_8990_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_8991_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_8992_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_8993_del__single__cont,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X4 = Mi )
        & ( X4 = 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 ) @ X4 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_8994_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_8995_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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 ) @ X4 ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_8996_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_8997_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_8998_succ__min,axiom,
    ! [Deg: nat,X4: 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 @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_8999_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X4: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X4 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_9000_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_9001_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L3: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L3 ) ) ) ).

% bit_concat_def
thf(fact_9002_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_9003_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_9004_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_9005_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_9006_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_9007_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_9008_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_9009_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_9010_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_9011_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_9012_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_9013_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_9014_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_9015_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_9016_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_9017_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_9018_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X4: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X4 @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_9019_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_9020_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_9021_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_9022_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_9023_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_9024_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_9025_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_9026_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_9027_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_9028_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_9029_le__num__One__iff,axiom,
    ! [X4: num] :
      ( ( ord_less_eq_num @ X4 @ one )
      = ( X4 = one ) ) ).

% le_num_One_iff
thf(fact_9030_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_9031_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_9032_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_9033_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_9034_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_9035_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_9036_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_9037_realpow__square__minus__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_9038_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_9039_nmod2,axiom,
    ! [N: int] :
      ( ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_9040_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_9041_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_9042_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_9043_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_9044_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_9045_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_9046_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_9047_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_9048_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_9049_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_9050_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_9051_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( 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_9052_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X4: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ ( 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_9053_nat__power__less__diff,axiom,
    ! [N: nat,Q5: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Q5 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% nat_power_less_diff
thf(fact_9054_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_9055_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_9056_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_9057_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_9058_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_9059_axxmod2,axiom,
    ! [X4: int] :
      ( ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X4 ) @ X4 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int )
      & ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X4 ) @ X4 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% axxmod2
thf(fact_9060_axxdiv2,axiom,
    ! [X4: int] :
      ( ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X4 ) @ X4 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X4 )
      & ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X4 ) @ X4 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X4 ) ) ).

% axxdiv2
thf(fact_9061_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_9062_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_9063_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_9064_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_9065_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_9066_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_9067_less__two__pow__divD,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( 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 @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_9068_less__two__pow__divI,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_nat @ X4 @ ( 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_9069_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_9070_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9071_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_9072_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_9073_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_9074_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_9075_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_9076_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_9077_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_9078_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_9079_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_9080_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_9081_exp__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_9082_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_9083_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
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_9084_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_9085_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_9086_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9087_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_9088_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_9089_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_9090_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_9091_ln__one__plus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_9092_real__exp__bound__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_9093_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q5: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_9094_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_9095_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_9096_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_9097_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( 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_9098_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_9099_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9100_exp__lower__Taylor__quadratic,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( divide_divide_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X4 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_9101_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_9102_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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_9103_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q5: int,R3: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q5 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q5 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_9104_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9105_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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_9106_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_9107_ln__one__minus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_9108_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( 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 @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_9109_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_9110_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_9111_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_9112_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_9113_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_9114_max__enat__simps_I3_J,axiom,
    ! [Q5: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q5 )
      = Q5 ) ).

% max_enat_simps(3)
thf(fact_9115_max__enat__simps_I2_J,axiom,
    ! [Q5: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q5 @ zero_z5237406670263579293d_enat )
      = Q5 ) ).

% max_enat_simps(2)
thf(fact_9116_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_9117_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_9118_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_9119_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_9120_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_9121_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_9122_div__half__nat,axiom,
    ! [Y: nat,X4: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X4 @ Y ) @ ( modulo_modulo_nat @ X4 @ Y ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y @ ( minus_minus_nat @ X4 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X4 @ ( 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 @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X4 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X4 @ ( 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 @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ ( minus_minus_nat @ X4 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).

% div_half_nat
thf(fact_9123_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X4 )
      @ ^ [R2: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R2
              = ( vEBT_vebt_member @ T @ X4 ) ) ) )
      @ ( 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_9124_vebt__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_buildupi_rule
thf(fact_9125_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_9126_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_9127_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_9128_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_9129_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_9130_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_9131_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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 ) @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_9132_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_9133_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_9134_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X4 ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_9135_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_9136_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_9137_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_9138_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X4 ) @ ( 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_9139_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_9140_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_9141_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_9142_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X4 ) @ ( 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_9143_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_9144_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_9145_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_9146_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_9147_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_9148_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_9149_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_9150_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_9151_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X4: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X4 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X4 ) ) @ ( 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_9152_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_9153_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_9154_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_9155_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_9156_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_9157_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_9158_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_9159_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_9160_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_9161_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_9162_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_9163_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_9164_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_9165_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_9166_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_9167_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_9168_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_9169_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_9170_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_9171_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_9172_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_9173_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_9174_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9175_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_9176_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_9177_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_9178_small__powers__of__2,axiom,
    ! [X4: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X4 )
     => ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X4 @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_9179_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X4 ) @ ( 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_9180_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X4 ) @ ( 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_9181_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_9182_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_9183_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_9184_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_9185_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_9186_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X: code_integer] : ( plus_p5714425477246183910nteger @ X @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9187_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9188_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_9189_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_9190_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_9191_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X4 )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( 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_9192_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_9193_VEBT__internal_Ospace_Oelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X4 )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X4
                = ( 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_9194_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_9195_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_9196_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_9197_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_9198_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_9199_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_9200_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_9201_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_9202_VEBT__internal_Ospace_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( 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_9203_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X4: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X4 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X4 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( 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_9204_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L2: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L2 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_9205_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ S2 @ 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 @ S2 ) )
                = ( 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 @ S2 @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9206_pred__list__to__short,axiom,
    ! [Deg: nat,X4: 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 @ X4 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X4 @ ( 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 ) @ X4 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_9207_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X4: 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 @ X4 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X4 @ ( 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 ) @ X4 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_9208_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N4: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% high_def
thf(fact_9209_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_9210_high__inv,axiom,
    ! [X4: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ ( 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 ) ) @ X4 ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_9211_log__ceil__idem,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_9212_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_9213_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9214_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_9215_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_9216_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9217_zero__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_9218_log__less__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_9219_one__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ A @ X4 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_9220_log__less__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_real @ X4 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_9221_log__less__cancel__iff,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X4 @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_9222_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_9223_zero__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_9224_log__le__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_9225_one__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ A @ X4 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_9226_log__le__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_eq_real @ X4 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_9227_log__le__cancel__iff,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_9228_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_9229_log__base__change,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X4 )
          = ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_9230_log__mult,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X4 @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_9231_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_9232_log__divide,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X4 @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_9233_log__base__pow,axiom,
    ! [A: real,N: nat,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X4 )
        = ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_9234_log__nat__power,axiom,
    ! [X4: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ B @ ( power_power_real @ X4 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X4 ) ) ) ) ).

% log_nat_power
thf(fact_9235_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_9236_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X4: 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 @ X4 )
             => ( ( log @ A @ X4 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X4 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_9237_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( 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 @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_9238_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_9239_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_9240_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_9241_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_9242_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9243_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9244_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_9245_log__base__10__eq2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% log_base_10_eq2
thf(fact_9246_log__base__10__eq1,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( 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 @ X4 ) ) ) ) ).

% log_base_10_eq1
thf(fact_9247_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9248_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9249_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_9250_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_9251_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_9252_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X4 )
        @ ^ [R2: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R2
                = ( vEBT_vebt_member @ T @ X4 ) ) ) )
        @ ( 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_9253_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X4 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X4 ) ) @ ( 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_9254_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_9255_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_9256_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9257_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9258_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9259_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9260_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_9261_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9262_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_9263_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_9264_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9265_nat__ceiling__le__eq,axiom,
    ! [X4: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) @ A )
      = ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9266_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_9267_numeral__power__less__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9268_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9269_numeral__power__le__nat__cancel__iff,axiom,
    ! [X4: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9270_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9271_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9272_nat__mono,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_9273_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_9274_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P2 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_9275_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P2 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_9276_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_9277_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_9278_nat__le__iff,axiom,
    ! [X4: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X4 ) @ N )
      = ( ord_less_eq_int @ X4 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_9279_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9280_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_9281_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9282_real__nat__ceiling__ge,axiom,
    ! [X4: real] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9283_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9284_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9285_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9286_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9287_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9288_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_9289_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_9290_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_9291_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N4: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ( P @ N4 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9292_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_9293_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_9294_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_9295_Suc__as__int,axiom,
    ( suc
    = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9296_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_9297_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9298_nat__diff__distrib_H,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X4 @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9299_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_9300_nat__div__distrib,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_9301_nat__div__distrib_H,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_9302_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9303_nat__mod__distrib,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X4 @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9304_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9305_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_9306_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_9307_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_9308_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_9309_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_9310_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_9311_pred__less__length__list,axiom,
    ! [Deg: nat,X4: 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 @ X4 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 ) @ X4 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X4 ) @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_9312_pred__lesseq__max,axiom,
    ! [Deg: nat,X4: 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 @ X4 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X4 ) @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_9313_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X4: 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 @ X4 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_9314_bit__split__inv,axiom,
    ! [X4: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X4 @ D ) @ ( vEBT_VEBT_low @ X4 @ D ) @ D )
      = X4 ) ).

% bit_split_inv
thf(fact_9315_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N4: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% low_def
thf(fact_9316_low__inv,axiom,
    ! [X4: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ ( 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 ) ) @ X4 ) @ N )
        = X4 ) ) ).

% low_inv
thf(fact_9317_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X4 ) ) ) ) ).

% both_member_options_ding
thf(fact_9318_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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 ) @ X4 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X4 = Mi )
          | ( X4 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_9319_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X4 = Mi )
          | ( X4 = Ma )
          | ( ( ord_less_nat @ X4 @ Ma )
            & ( ord_less_nat @ Mi @ X4 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_9320_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X4: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( 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 ) @ X4 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_9321_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X4: 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 @ X4 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X4 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X4 @ ( 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_9322_insert__simp__norm,axiom,
    ! [X4: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X4 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X4 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X4 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X4 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_9323_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X4: 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 @ X4 )
        & ( ord_less_eq_nat @ X4 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X4 @ ( 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 @ X4 @ ( 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 ) @ X4 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X4 = 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_9324_del__x__mi__lets__in__not__minNull,axiom,
    ! [X4: 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] :
      ( ( ( X4 = Mi )
        & ( ord_less_nat @ X4 @ 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 ) @ X4 )
                          = ( 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_9325_del__x__not__mia,axiom,
    ! [Mi: nat,X4: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X4 )
        & ( ord_less_eq_nat @ X4 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 ) @ X4 )
                  = ( 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 @ ( X4 = 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 @ ( X4 = 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_9326_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X4: 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 @ X4 )
        & ( ord_less_eq_nat @ X4 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X4 @ ( 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 @ X4 @ ( 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 ) @ X4 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X4 = 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_9327_del__x__not__mi,axiom,
    ! [Mi: nat,X4: 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 @ X4 )
        & ( ord_less_eq_nat @ X4 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X4 @ ( 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 @ X4 @ ( 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 ) @ X4 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X4 = 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 ) @ X4 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X4 = 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_9328_del__x__mia,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X4 = Mi )
        & ( ord_less_nat @ X4 @ 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 ) @ X4 )
            = ( 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_9329_del__x__mi__lets__in__minNull,axiom,
    ! [X4: 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] :
      ( ( ( X4 = Mi )
        & ( ord_less_nat @ X4 @ 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 ) @ X4 )
                            = ( 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_9330_del__x__mi__lets__in,axiom,
    ! [X4: 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] :
      ( ( ( X4 = Mi )
        & ( ord_less_nat @ X4 @ 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 ) @ X4 )
                            = ( 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 ) @ X4 )
                            = ( 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_9331_del__x__mi,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X4 = Mi )
        & ( ord_less_nat @ X4 @ 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 ) @ X4 )
                    = ( 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_9332_del__in__range,axiom,
    ! [Mi: nat,X4: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X4 )
        & ( ord_less_eq_nat @ X4 @ 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 ) @ X4 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X4 = 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 ) )
                          & ( ( X4 != Mi )
                           => ( X4 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X4 = 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 ) )
                          & ( ( X4 != Mi )
                           => ( X4 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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_9333_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X4: 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 @ X4 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 ) @ X4 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_9334_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X4: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X4 @ ( 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 @ X4 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_9335_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S2 ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_9336_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_9337_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
      = ( ( X4 != Mi )
       => ( ( X4 != Ma )
         => ( ~ ( ord_less_nat @ X4 @ Mi )
            & ( ~ ( ord_less_nat @ X4 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X4 )
                & ( ~ ( ord_less_nat @ Ma @ X4 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_9338_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X4 )
      = ( ( X4 = Mi )
        | ( X4 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_9339_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_9340_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A4 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B3 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_9341_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A4 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B3 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_9342_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_9343_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_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [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_9344_vebt__member_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X4 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A4 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B3 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_9345_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X4
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_9346_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_9347_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,X4: 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 ) @ X4 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X4 = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X4 = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X4 ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X4 )
                    & ( ord_less_nat @ X4 @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( 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_9348_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,X4: 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 ) @ X4 )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X4 = Mi )
              | ( X4 = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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_9349_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X4 = Mi )
              | ( X4 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ X4 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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_9350_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_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [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_9351_vebt__member_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X4 @ Xa )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_9352_vebt__member_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X4 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A4 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B3 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X4
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X4
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_9353_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X4
                        = ( 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 @ ( Xa = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa )
                                    & ( ord_less_nat @ Xa @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_9354_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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_9355_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A4: $o,B3: $o] :
              ( A1
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X6 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                     => ( ( M2 = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M2 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X6: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X6 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                       => ( ( M2
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M2 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X6 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                         => ( ( M2 = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M2 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X6: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X6: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X6 )
                                                      & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X6: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X6 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                           => ( ( M2
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M2 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X6: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X6: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X6 )
                                                        & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_9356_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A2: $o,B2: $o] :
                ( A12
                = ( vEBT_Leaf @ A2 @ B2 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N4 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N4 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_9357_vebt__insert_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X4 @ Xa )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_9358_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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_9359_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X4: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X4 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X4 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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_9360_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,
    ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X4 @ Mi )
          | ( ord_less_nat @ Ma @ X4 ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X4 @ Mi )
            | ( ord_less_nat @ Ma @ X4 ) )
       => ( ( ( ( X4 = Mi )
              & ( X4 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
              = one_one_nat ) )
          & ( ~ ( ( X4 = Mi )
                & ( X4 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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_9361_vebt__succ_Osimps_I6_J,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X4 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_9362_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X4: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X4 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X4 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X4 ) @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_9363_vebt__delete_Osimps_I7_J,axiom,
    ! [X4: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X4 @ Mi )
          | ( ord_less_nat @ Ma @ X4 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X4 @ Mi )
            | ( ord_less_nat @ Ma @ X4 ) )
       => ( ( ( ( X4 = Mi )
              & ( X4 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X4 = Mi )
                & ( X4 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X4 = 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 ) )
                            & ( ( X4 != Mi )
                             => ( X4 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X4 = Mi ) @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X4 = 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 ) )
                            & ( ( X4 != Mi )
                             => ( X4 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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_9364_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X4
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N2: nat] :
                  ( Xa
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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_9365_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] :
                ( X4
                = ( vEBT_Leaf @ A4 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [Va: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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_9366_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,X4: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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_9367_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,X4: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X4 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X4 ) @ 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 @ X4 @ ( 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 @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( 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_9368_vebt__delete_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X4 @ Xa )
        = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A4: $o] :
              ( ? [B3: $o] :
                  ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A4 @ $false ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [N2: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A4 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X4
                    = ( 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] :
                    ( ( X4
                      = ( 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] :
                      ( ( X4
                        = ( 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] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_9369_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B3: $o] :
                ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [N2: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X4
                      = ( 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] :
                        ( X4
                        = ( 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] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_9370_vebt__succ_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X4 @ Xa )
        = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( X4
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X4
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N2: nat] :
                  ( Xa
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_9371_vebt__pred_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] :
                  ( X4
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A4
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A4
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [Va: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A4
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A4
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_9372_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X4
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N2: nat] :
                  ( Xa
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ Xa @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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_9373_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] :
                ( X4
                = ( vEBT_Leaf @ A4 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [Va: nat] :
                      ( Xa
                      = ( 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 ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( 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 @ Xa ) @ one_one_nat
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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_9374_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,X4: 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 ) @ X4 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X4 @ Mi )
            | ( ord_less_nat @ Ma @ X4 ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X4 = Mi )
                & ( X4 = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X4 = 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( if_nat
                              @ ( ( ( X4 = 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 ) )
                                & ( ( X4 != Mi )
                                 => ( X4 = 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( if_nat
                            @ ( ( ( X4 = 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 ) )
                              & ( ( X4 != Mi )
                               => ( X4 = 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 @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X4 = 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 ) ) ) ) ) ) @ X4 ) @ ( 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_9375_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B3: $o] :
                ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [N2: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X4
                      = ( 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] :
                        ( X4
                        = ( 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] :
                        ( ( X4
                          = ( 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 @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_9376_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N4: nat,TreeList4: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X @ N4 ) ) @ ( vEBT_VEBT_low @ X @ N4 ) ) ) ) ).

% in_children_def
thf(fact_9377_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( 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 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( 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 @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                        @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa = Mi2 )
                                                         => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa = Mi2 )
                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa != Mi2 )
                                                       => ( Xa = Ma2 ) ) )
                                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_9378_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = 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 @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N2: nat] :
                    ( ( Xa
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( 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 @ Xa @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_9379_vebt__succ_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = 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 @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N2: nat] :
                    ( ( Xa
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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 @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_9380_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( 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 @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa
                        = ( 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 @ A4 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( 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 @ Xa ) @ one_one_nat
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_9381_vebt__pred_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A4
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A4
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B3
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A4
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A4
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_9382_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( 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 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_9383_vebt__delete_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A4 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A4 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X4
                          = ( 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 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_9384_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N2: nat] :
                    ( ( Xa
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_9385_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X4
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X4
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X4
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X4
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_9386_vebt__insert_Opelims,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_9387_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_9388_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ ( Xa = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa )
                                        & ( ord_less_nat @ Xa @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_9389_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( 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] :
                        ( X4
                        = ( 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 @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_9390_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X4 @ Xa )
        = Y )
     => ( ( ? [A4: $o,B3: $o] :
              ( X4
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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_9391_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,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_9392_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,S2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X4 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_9393_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,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X4 = 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_9394_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X4 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_9395_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X4 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_9396_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_9397_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_9398_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X4 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_9399_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_9400_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,X4: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X4 = 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_9401_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X4 ) @ ( 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_9402_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X4: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X4 ) @ ( 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_9403_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9404_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X4: 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 @ X4 ) ) @ ( 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_9405_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,X4: 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 ) @ X4 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X4 = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X4 = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X4 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( 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 @ X4 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( 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_9406_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,X4: 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 ) @ X4 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X4 = Mi )
                | ( X4 = 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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 @ X4 @ Mi ) @ Mi @ X4 ) @ ( 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_9407_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_9408_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,
    ! [X4: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( 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 @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_9409_vebt__member_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A4 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B3 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_9410_vebt__member_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X4
                  = ( 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 ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_9411_vebt__member_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_9412_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_9413_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A4 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_9414_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A4 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B3 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_9415_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_9416_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( 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 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_9417_lowi__h,axiom,
    ! [X4: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X4 @ N )
      @ ^ [R2: nat] :
          ( pure_assn
          @ ( R2
            = ( vEBT_VEBT_low @ X4 @ N ) ) ) ) ).

% lowi_h
thf(fact_9418_TBOUND__lowi,axiom,
    ! [X4: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X4 @ N ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_9419_lowi__hT,axiom,
    ! [X4: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X4 @ N )
      @ ^ [R2: nat] :
          ( pure_assn
          @ ( R2
            = ( vEBT_VEBT_low @ X4 @ N ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_9420_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X4
                = ( 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 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_9421_highi__h,axiom,
    ! [X4: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X4 @ N )
      @ ^ [R2: nat] :
          ( pure_assn
          @ ( R2
            = ( vEBT_VEBT_high @ X4 @ N ) ) ) ) ).

% highi_h
thf(fact_9422_highi__hT,axiom,
    ! [X4: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X4 @ N )
      @ ^ [R2: nat] :
          ( pure_assn
          @ ( R2
            = ( vEBT_VEBT_high @ X4 @ N ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_9423_monoseq__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_9424_TBOUND__highi,axiom,
    ! [X4: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X4 @ N ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_9425_monoseq__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X4 ) ) ) ) ).

% monoseq_realpow
thf(fact_9426_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ N @ A3 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).

% set_encode_insert
thf(fact_9427_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_9428_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_9429_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_9430_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_9431_set__encode__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( nat_set_encode @ A3 )
            = ( nat_set_encode @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% set_encode_eq
thf(fact_9432_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( nat_set_encode @ A3 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_9433_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_9434_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_9435_ln__series,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X4 )
          = ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_9436_arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arctan @ X4 )
        = ( 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 @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_9437_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_9438_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_9439_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_9440_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_9441_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_9442_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_9443_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_9444_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_9445_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_9446_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_9447_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_9448_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_9449_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_9450_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_9451_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_9452_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_9453_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_9454_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_9455_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_9456_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_9457_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_9458_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_9459_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_9460_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_9461_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_9462_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X4: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_9463_bezout__add__nat,axiom,
    ! [A: nat,B: 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 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) ) ) ).

% bezout_add_nat
thf(fact_9464_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D2 @ A )
      & ( dvd_dvd_nat @ D2 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D2 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D2 ) ) ) ).

% bezout1_nat
thf(fact_9465_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_9466_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_9467_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_9468_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_9469_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_9470_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q5: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q5 )
          = ( modulo_modulo_nat @ N @ Q5 ) )
        = ( dvd_dvd_nat @ Q5 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_9471_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_9472_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_9473_div2__even__ext__nat,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X4 = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_9474_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_9475_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_9476_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_9477_dvd__minus__add,axiom,
    ! [Q5: nat,N: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q5 @ N )
     => ( ( ord_less_eq_nat @ Q5 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q5 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q5 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_9478_mod__nat__eqI,axiom,
    ! [R3: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_9479_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_9480_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_9481_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_9482_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_9483_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).

% even_set_encode_iff
thf(fact_9484_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_9485_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_9486_Bernoulli__inequality__even,axiom,
    ! [N: nat,X4: 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 ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_9487_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_9488_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_9489_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_9490_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N2: nat] :
                ( ( X4
                  = ( 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_9491_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_9492_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X4
                  = ( 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_9493_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_9494_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_9495_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_9496_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_9497_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_9498_vebt__buildup_Oelims,axiom,
    ! [X4: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X4
                  = ( 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_9499_VEBT__internal_OTb_Oelims,axiom,
    ! [X4: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X4
                  = ( 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_9500_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,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X4
                  = ( 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_9501_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,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X4
                  = ( 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_9502_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X4: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( X4
                  = ( 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_9503_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X4: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X4
                    = ( 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_9504_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,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X4
                    = ( 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_9505_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,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X4
                    = ( 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_9506_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ N ) ) )
        = ( abs_abs_int @ N ) ) ) ).

% Max_divisors_self_int
thf(fact_9507_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R3 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9508_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9509_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R3 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9510_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9511_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_9512_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_9513_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_9514_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_9515_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_9516_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_9517_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_9518_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9519_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_9520_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_9521_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_9522_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_9523_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_9524_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_9525_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_9526_vebt__buildup_Opelims,axiom,
    ! [X4: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X4
                    = ( 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_9527_VEBT__internal_OTb_Opelims,axiom,
    ! [X4: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X4
                    = ( 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_9528_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X4
                    = ( 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_9529_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X4
                    = ( 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_9530_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_9531_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_9532_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_9533_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_9534_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_9535_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_9536_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_9537_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_9538_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_9539_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_9540_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_9541_arctan__inverse,axiom,
    ! [X4: real] :
      ( ( X4 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X4 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X4 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X4 ) ) ) ) ).

% arctan_inverse
thf(fact_9542_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X4 @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9543_Maclaurin__cos__expansion2,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X4 )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9544_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X4 )
            & ( ( sin_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9545_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_9546_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_9547_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_9548_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_9549_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_9550_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_9551_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_9552_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_9553_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_9554_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_9555_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_9556_sincos__principal__value,axiom,
    ! [X4: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X4 ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X4 ) ) ) ).

% sincos_principal_value
thf(fact_9557_sin__zero__abs__cos__one,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_9558_sin__x__le__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( sin_real @ X4 ) @ X4 ) ) ).

% sin_x_le_x
thf(fact_9559_sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sin_real @ X4 ) @ one_one_real ) ).

% sin_le_one
thf(fact_9560_cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( cos_real @ X4 ) @ one_one_real ) ).

% cos_le_one
thf(fact_9561_abs__sin__x__le__abs__x,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ ( abs_abs_real @ X4 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_9562_cos__arctan__not__zero,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( arctan @ X4 ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_9563_sin__cos__le1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_9564_sin__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero
thf(fact_9565_sin__x__ge__neg__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ ( sin_real @ X4 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_9566_sin__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_ge_zero
thf(fact_9567_sin__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X4 ) ) ).

% sin_ge_minus_one
thf(fact_9568_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_9569_cos__mono__le__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_9570_cos__inj__pi,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X4 )
                = ( cos_real @ Y ) )
             => ( X4 = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_9571_cos__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X4 ) ) ).

% cos_ge_minus_one
thf(fact_9572_abs__sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_9573_abs__cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X4 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_9574_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_9575_cos__mono__less__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X4 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_9576_cos__monotone__0__pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_9577_sin__eq__0__pi,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ( ( sin_real @ X4 )
            = zero_zero_real )
         => ( X4 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_9578_sin__zero__pi__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ pi )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
        = ( X4 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_9579_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_9580_sin__zero__iff__int2,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X4
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_9581_sincos__total__pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( 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 )
            & ( X4
              = ( cos_real @ T6 ) )
            & ( Y
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_9582_sin__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_9583_cos__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_9584_sin__gt__zero__02,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_9585_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_9586_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_9587_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_9588_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_9589_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_9590_sincos__total__pi__half,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( 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 ) ) ) )
              & ( X4
                = ( cos_real @ T6 ) )
              & ( Y
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_9591_sincos__total__2pi__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( 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 ) )
          & ( X4
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_9592_sincos__total__2pi,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( 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 ) )
             => ( ( X4
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_9593_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_9594_sin__gt__zero2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero2
thf(fact_9595_sin__lt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_9596_cos__double__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_9597_cos__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero
thf(fact_9598_sin__monotone__2pi__le,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_9599_sin__mono__le__eq,axiom,
    ! [X4: real,Y: 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 ) ) ) )
       => ( ( 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 @ X4 ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_9600_sin__inj__pi,axiom,
    ! [X4: real,Y: 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 ) ) ) )
       => ( ( 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 @ X4 )
                = ( sin_real @ Y ) )
             => ( X4 = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_9601_sin__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_9602_sin__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_9603_sin__mono__less__eq,axiom,
    ! [X4: real,Y: 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 ) ) ) )
       => ( ( 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 @ X4 ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X4 @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_9604_sin__monotone__2pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_9605_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_9606_cos__gt__zero__pi,axiom,
    ! [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 ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_9607_cos__ge__zero,axiom,
    ! [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 ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_ge_zero
thf(fact_9608_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_9609_sin__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_9610_cos__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_9611_sin__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_9612_sin__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_9613_cos__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( cos_real @ X4 )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_9614_cos__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_9615_Maclaurin__sin__expansion,axiom,
    ! [X4: real,N: nat] :
    ? [T6: real] :
      ( ( sin_real @ X4 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9616_Maclaurin__sin__expansion2,axiom,
    ! [X4: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X4 ) )
      & ( ( sin_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9617_Maclaurin__cos__expansion,axiom,
    ! [X4: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X4 ) )
      & ( ( cos_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9618_Maclaurin__sin__expansion4,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X4 )
          & ( ( sin_real @ X4 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X4 @ 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 @ X4 @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9619_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X: nat,N4: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% lowi_def
thf(fact_9620_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X: nat,N4: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% highi_def
thf(fact_9621_highsimp,axiom,
    ! [X4: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X4 @ N ) )
      = ( vEBT_VEBT_highi @ X4 @ N ) ) ).

% highsimp
thf(fact_9622_lowsimp,axiom,
    ! [X4: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X4 @ N ) )
      = ( vEBT_VEBT_lowi @ X4 @ N ) ) ).

% lowsimp
thf(fact_9623_summable__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ 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 @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_9624_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_9625_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_9626_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_9627_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_9628_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
     => ( summable_real @ F ) ) ).

% summable_rabs_cancel
thf(fact_9629_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
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_9630_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs
thf(fact_9631_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_9632_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_9633_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_9634_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_9635_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_9636_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_9637_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_9638_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_9639_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_9640_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_9641_tanh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( tanh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_9642_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_9643_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_9644_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_9645_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_9646_tanh__real__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( tanh_real @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_9647_tanh__real__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% tanh_real_le_iff
thf(fact_9648_tanh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_pos_iff
thf(fact_9649_tanh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_9650_tanh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_9651_tanh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_nonneg_iff
thf(fact_9652_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_9653_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_9654_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_9655_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_9656_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_9657_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( A
          = ( numeral_numeral_real @ W2 ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_9658_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_9659_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_9660_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_9661_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_9662_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_9663_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9664_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9665_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9666_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_9667_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9668_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9669_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_9670_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_9671_tan__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_gt_zero
thf(fact_9672_tan__pos__pi2__le,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_9673_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_9674_tan__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X4 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_9675_tan__mono__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_9676_tan__mono__le__eq,axiom,
    ! [X4: real,Y: 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 ) ) ) )
       => ( ( 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 @ X4 ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_9677_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_9678_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9679_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9680_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_9681_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_9682_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9683_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_9684_sin__paired,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
      @ ( sin_real @ X4 ) ) ).

% sin_paired
thf(fact_9685_ceiling__log__eq__powr__iff,axiom,
    ! [X4: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X4 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X4 )
            & ( ord_less_eq_real @ X4 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_9686_powr__gt__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X4 @ A ) )
      = ( X4 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_9687_powr__nonneg__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X4 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_9688_powr__eq__one__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X4 )
          = one_one_real )
        = ( X4 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_9689_powr__one__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( powr_real @ X4 @ one_one_real )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% powr_one_gt_zero_iff
thf(fact_9690_powr__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ one_one_real )
        = X4 ) ) ).

% powr_one
thf(fact_9691_powr__le__cancel__iff,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_9692_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_9693_powr__log__cancel,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( powr_real @ A @ ( log @ A @ X4 ) )
            = X4 ) ) ) ) ).

% powr_log_cancel
thf(fact_9694_powr__numeral,axiom,
    ! [X4: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_9695_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( ord_less_eq_int @ Xa @ X4 ) ) ).

% less_eq_integer.abs_eq
thf(fact_9696_powr__mono,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X4 )
       => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) ) ) ) ).

% powr_mono
thf(fact_9697_powr__mono2,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_9698_powr__ge__pzero,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X4 @ Y ) ) ).

% powr_ge_pzero
thf(fact_9699_powr__less__mono2__neg,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_9700_powr__non__neg,axiom,
    ! [A: real,X4: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X4 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_9701_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9702_powr__mono2_H,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_9703_powr__less__mono2,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_9704_powr__inj,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X4 )
            = ( powr_real @ A @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% powr_inj
thf(fact_9705_gr__one__powr,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X4 @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_9706_ge__one__powr__ge__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_9707_powr__mono__both,axiom,
    ! [A: real,B: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X4 )
         => ( ( ord_less_eq_real @ X4 @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_9708_powr__le1,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_9709_powr__divide,axiom,
    ! [X4: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X4 @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_9710_powr__mult,axiom,
    ! [X4: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X4 @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_9711_log__base__powr,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X4 )
        = ( divide_divide_real @ ( log @ A @ X4 ) @ B ) ) ) ).

% log_base_powr
thf(fact_9712_log__powr,axiom,
    ! [X4: real,B: real,Y: real] :
      ( ( X4 != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X4 @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X4 ) ) ) ) ).

% log_powr
thf(fact_9713_ln__powr,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X4 @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_powr
thf(fact_9714_powr__realpow,axiom,
    ! [X4: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X4 @ N ) ) ) ).

% powr_realpow
thf(fact_9715_less__log__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ Y @ ( log @ B @ X4 ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ).

% less_log_iff
thf(fact_9716_log__less__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ B @ X4 ) @ Y )
          = ( ord_less_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_9717_less__powr__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X4 ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_9718_powr__less__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X4 )
          = ( ord_less_real @ Y @ ( log @ B @ X4 ) ) ) ) ) ).

% powr_less_iff
thf(fact_9719_powr__neg__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X4 ) ) ) ).

% powr_neg_one
thf(fact_9720_powr__mult__base,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ X4 @ ( powr_real @ X4 @ Y ) )
        = ( powr_real @ X4 @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_9721_powr__le__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X4 )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X4 ) ) ) ) ) ).

% powr_le_iff
thf(fact_9722_le__powr__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X4 ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_9723_log__le__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ B @ X4 ) @ Y )
          = ( ord_less_eq_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_9724_le__log__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X4 ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ).

% le_log_iff
thf(fact_9725_ln__powr__bound,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( divide_divide_real @ ( powr_real @ X4 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_9726_ln__powr__bound2,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X4 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X4 ) ) ) ) ).

% ln_powr_bound2
thf(fact_9727_add__log__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X4 ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_9728_log__add__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ ( log @ B @ X4 ) @ Y )
            = ( log @ B @ ( times_times_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_9729_minus__log__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X4 ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_9730_power__half__series,axiom,
    ( sums_real
    @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_9731_log__minus__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ ( log @ B @ X4 ) @ Y )
            = ( log @ B @ ( times_times_real @ X4 @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_9732_sums__if_H,axiom,
    ! [G: nat > real,X4: real] :
      ( ( sums_real @ G @ X4 )
     => ( sums_real
        @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X4 ) ) ).

% sums_if'
thf(fact_9733_sums__if,axiom,
    ! [G: nat > real,X4: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X4 )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X4 @ Y ) ) ) ) ).

% sums_if
thf(fact_9734_powr__neg__numeral,axiom,
    ! [X4: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_9735_powr__int,axiom,
    ! [X4: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X4 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9736_cos__paired,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
      @ ( cos_real @ X4 ) ) ).

% cos_paired
thf(fact_9737_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_9738_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_9739_nat__floor__neg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9740_floor__eq3,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9741_le__nat__floor,axiom,
    ! [X4: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ A )
     => ( ord_less_eq_nat @ X4 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9742_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_9743_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P @ D2 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9744_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N4: nat] :
            ( ( N4 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_9745_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P @ D2 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_9746_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_9747_ln__inverse,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_inverse
thf(fact_9748_floor__eq4,axiom,
    ! [N: nat,X4: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9749_floor__eq2,axiom,
    ! [N: int,X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = N ) ) ) ).

% floor_eq2
thf(fact_9750_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_9751_log__inverse,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( log @ A @ ( inverse_inverse_real @ X4 ) )
            = ( uminus_uminus_real @ ( log @ A @ X4 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9752_exp__plus__inverse__exp,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9753_plus__inverse__ge__2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9754_real__le__x__sinh,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ X4 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9755_real__le__abs__sinh,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9756_floor__log__eq__powr__iff,axiom,
    ! [X4: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X4 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X4 )
            & ( ord_less_real @ X4 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9757_powr__real__of__int,axiom,
    ! [X4: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X4 @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9758_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_9759_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_9760_Maclaurin__sin__bound,axiom,
    ! [X4: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X4 )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X4 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X4 ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9761_sinh__real__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( sinh_real @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9762_sinh__real__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% sinh_real_le_iff
thf(fact_9763_sinh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9764_sinh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_pos_iff
thf(fact_9765_sinh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9766_sinh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_nonneg_iff
thf(fact_9767_cosh__real__ge__1,axiom,
    ! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_ge_1
thf(fact_9768_cosh__real__nonzero,axiom,
    ! [X4: real] :
      ( ( cosh_real @ X4 )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9769_sinh__le__cosh__real,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).

% sinh_le_cosh_real
thf(fact_9770_cosh__real__nonneg,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_nonneg
thf(fact_9771_cosh__real__nonneg__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9772_cosh__real__nonpos__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9773_arcosh__cosh__real,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( arcosh_real @ ( cosh_real @ X4 ) )
        = X4 ) ) ).

% arcosh_cosh_real
thf(fact_9774_cosh__real__pos,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_pos
thf(fact_9775_cosh__real__strict__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9776_cosh__real__nonneg__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9777_cosh__real__nonpos__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X4 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9778_cosh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( cosh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9779_sinh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( sinh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9780_cot__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X4 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9781_le__arcsin__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ 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 @ X4 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X4 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_9782_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_9783_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9784_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_9785_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9786_arcsin__le__arcsin,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9787_arcsin__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X4 ) )
          = ( uminus_uminus_real @ ( arcsin @ X4 ) ) ) ) ) ).

% arcsin_minus
thf(fact_9788_arcsin__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_9789_arcsin__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X4 )
            = ( arcsin @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9790_arcsin__less__arcsin,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9791_arcsin__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_9792_cos__arcsin__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9793_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_9794_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_9795_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_9796_cot__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X4 ) ) ) ) ).

% cot_gt_zero
thf(fact_9797_arcsin__sin,axiom,
    ! [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 ) ) ) )
       => ( ( arcsin @ ( sin_real @ X4 ) )
          = X4 ) ) ) ).

% arcsin_sin
thf(fact_9798_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_9799_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_9800_arcsin__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ 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 @ X4 ) @ Y )
              = ( ord_less_eq_real @ X4 @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_9801_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_9802_set__decode__0,axiom,
    ! [X4: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X4 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) ) ) ).

% set_decode_0
thf(fact_9803_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_9804_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_9805_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_9806_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
        = A3 ) ) ).

% set_encode_inverse
thf(fact_9807_set__decode__Suc,axiom,
    ! [N: nat,X4: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X4 ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_9808_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_9809_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_9810_Complex__eq__i,axiom,
    ! [X4: real,Y: real] :
      ( ( ( complex2 @ X4 @ Y )
        = imaginary_unit )
      = ( ( X4 = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9811_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9812_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_9813_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_9814_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_9815_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9816_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_9817_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_9818_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9819_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L3 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9820_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N4: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_9821_and__int_Opelims,axiom,
    ! [X4: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_9822_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9823_and__int_Oelims,axiom,
    ! [X4: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y )
     => ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9824_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_9825_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_9826_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9827_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9828_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9829_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_9830_AND__upper2,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_9831_AND__upper1,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ X4 ) ) ).

% AND_upper1
thf(fact_9832_AND__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) ) ) ).

% AND_lower
thf(fact_9833_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9834_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_9835_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_9836_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L3 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L3
            @ ( if_int
              @ ( L3
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9837_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9838_Divides_Oadjust__div__eq,axiom,
    ! [Q5: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q5 @ R3 ) )
      = ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_9839_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P6: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_9840_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_9841_and__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9842_and__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9843_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_9844_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_9845_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_9846_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9847_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9848_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q7: nat,R2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R2 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ R2 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9849_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q7: int,R2: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R2 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ R2 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9850_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q7: code_integer,R2: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R2 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ R2 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9851_False__map2__and,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ $false @ bot_bot_set_o ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_o @ Xs2 ) )
       => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (&) ) @ ( zip_o_o @ Xs2 @ Ys ) )
          = Xs2 ) ) ) ).

% False_map2_and
thf(fact_9852_False__map2__or,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ $false @ bot_bot_set_o ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( size_size_list_o @ Xs2 ) )
       => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (|) ) @ ( zip_o_o @ Xs2 @ Ys ) )
          = Ys ) ) ) ).

% False_map2_or
thf(fact_9853_align__lem__or,axiom,
    ! [Xs2: list_o,N: nat,M: nat,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( plus_plus_nat @ N @ M ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( plus_plus_nat @ N @ M ) )
       => ( ( ( drop_o @ M @ Xs2 )
            = ( replicate_o @ N @ $false ) )
         => ( ( ( take_o @ M @ Ys )
              = ( replicate_o @ M @ $false ) )
           => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (|) ) @ ( zip_o_o @ Xs2 @ Ys ) )
              = ( append_o @ ( take_o @ M @ Xs2 ) @ ( drop_o @ M @ Ys ) ) ) ) ) ) ) ).

% align_lem_or
thf(fact_9854_align__lem__and,axiom,
    ! [Xs2: list_o,N: nat,M: nat,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( plus_plus_nat @ N @ M ) )
     => ( ( ( size_size_list_o @ Ys )
          = ( plus_plus_nat @ N @ M ) )
       => ( ( ( drop_o @ M @ Xs2 )
            = ( replicate_o @ N @ $false ) )
         => ( ( ( take_o @ M @ Ys )
              = ( replicate_o @ M @ $false ) )
           => ( ( map_Pr7541730621154948341_o_o_o @ ( produc6197397395684419436_o_o_o @ (&) ) @ ( zip_o_o @ Xs2 @ Ys ) )
              = ( replicate_o @ ( plus_plus_nat @ N @ M ) @ $false ) ) ) ) ) ) ).

% align_lem_and
thf(fact_9855_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M5: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N4 ) ) @ M5 ) ) ) ).

% prod_encode_def
thf(fact_9856_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q7: int,R2: int] : ( plus_plus_int @ Q7 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_9857_sort__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X: nat] : X
        @ ( upt @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sort_upt
thf(fact_9858_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X: int] : X
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_9859_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q7: nat] : ( product_Pair_nat_nat @ ( suc @ Q7 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9860_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_9861_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_9862_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_9863_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9864_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9865_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_9866_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_9867_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_9868_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_9869_arctan__def,axiom,
    ( arctan
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y4 ) ) ) ) ) ).

% arctan_def
thf(fact_9870_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y4 ) ) ) ) ) ).

% arcsin_def
thf(fact_9871_ln__real__def,axiom,
    ( ln_ln_real
    = ( ^ [X: real] :
          ( the_real
          @ ^ [U2: real] :
              ( ( exp_real @ U2 )
              = X ) ) ) ) ).

% ln_real_def
thf(fact_9872_ln__neg__is__const,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ln_ln_real @ X4 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9873_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9874_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9875_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9876_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_9877_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_9878_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_9879_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_9880_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_9881_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_9882_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q5 ) @ ( bit_se2925701944663578781it_nat @ N @ Q5 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9883_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_9884_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_9885_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_9886_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9887_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9888_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_9889_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_9890_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_9891_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_9892_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_9893_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_9894_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_9895_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_9896_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_9897_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_9898_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_9899_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9900_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_9901_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9902_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_9903_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_9904_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_9905_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y4: rat] :
          ( ( ord_less_rat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9906_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_rat_def
thf(fact_9907_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9908_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R3 )
     => ~ ! [S: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R3
                 != ( plus_plus_rat @ S @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9909_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_9910_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9911_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_9912_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_9913_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_9914_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_9915_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_9916_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_9917_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_9918_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_9919_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_9920_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_9921_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_9922_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_9923_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9924_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9925_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L3 )
          @ ( if_int
            @ ( L3
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9926_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_9927_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_9928_XOR__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X4 @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9929_int__xor__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ J )
      = J ) ).

% int_xor_code(1)
thf(fact_9930_int__xor__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se6526347334894502574or_int @ I @ zero_zero_int )
      = I ) ).

% int_xor_code(2)
thf(fact_9931_XOR__upper,axiom,
    ! [X4: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( 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 @ X4 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9932_int__and__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se725231765392027082nd_int @ I @ zero_zero_int )
      = zero_zero_int ) ).

% int_and_code(2)
thf(fact_9933_int__and__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ J )
      = zero_zero_int ) ).

% int_and_code(1)
thf(fact_9934_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9935_arcosh__real__def,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( arcosh_real @ X4 )
        = ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_9936_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X4: real] :
      ( ( ( sqrt @ X4 )
        = zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_9937_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_9938_real__sqrt__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_9939_real__sqrt__lt__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_9940_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_9941_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_9942_real__sqrt__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_9943_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_9944_real__sqrt__le__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_9945_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_9946_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_9947_xor__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9948_xor__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9949_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_9950_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_9951_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_9952_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_9953_real__sqrt__pow2__iff,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% real_sqrt_pow2_iff
thf(fact_9954_real__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 ) ) ).

% real_sqrt_pow2
thf(fact_9955_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_9956_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_9957_test__bit__int__code_I1_J,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ zero_zero_int @ N ) ).

% test_bit_int_code(1)
thf(fact_9958_real__sqrt__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_9959_real__sqrt__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_9960_real__sqrt__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_one
thf(fact_9961_real__sqrt__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_9962_real__sqrt__eq__zero__cancel,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sqrt @ X4 )
          = zero_zero_real )
       => ( X4 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_9963_real__div__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ X4 @ ( sqrt @ X4 ) )
        = ( sqrt @ X4 ) ) ) ).

% real_div_sqrt
thf(fact_9964_sqrt__add__le__add__sqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X4 @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_9965_le__real__sqrt__sumsq,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_9966_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_9967_sqrt__divide__self__eq,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ ( sqrt @ X4 ) @ X4 )
        = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9968_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_9969_sqrt__le__D,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y )
     => ( ord_less_eq_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_9970_real__le__rsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X4 @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_9971_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N2 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K @ M3 )
                = ( 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_9972_real__sqrt__unique,axiom,
    ! [Y: real,X4: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X4 )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_9973_real__le__lsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_9974_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_9975_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X4: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X4 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_9976_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X4: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X4 )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_9977_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_9978_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X4: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_9979_real__sqrt__sum__squares__ge1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_9980_sqrt__ge__absD,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_9981_real__less__lsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X4 ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_9982_sqrt__sum__squares__le__sum,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X4 @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_9983_real__inv__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X4 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9984_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_9985_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_9986_real__sqrt__ge__abs1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_9987_ln__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( sqrt @ X4 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_9988_real__sqrt__power__even,axiom,
    ! [N: nat,X4: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( sqrt @ X4 ) @ N )
          = ( power_power_real @ X4 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_9989_arsinh__real__aux,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_9990_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X4: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_9991_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = 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 @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9992_arith__geo__mean__sqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X4 @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_9993_powr__half__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X4 ) ) ) ).

% powr_half_sqrt
thf(fact_9994_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9995_cos__x__y__le__one,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_9996_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_9997_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_9998_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_9999_sqrt__sum__squares__half__less,axiom,
    ! [X4: real,U: real,Y: real] :
      ( ( ord_less_real @ X4 @ ( 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 @ X4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_10000_sin__cos__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) )
     => ( ( sin_real @ X4 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_10001_cos__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_10002_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_10003_sin__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_10004_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_10005_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_10006_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_10007_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_10008_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_10009_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_10010_arccos__le__arccos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_10011_arccos__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X4 ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ).

% arccos_le_mono
thf(fact_10012_arccos__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X4 )
          = ( arccos @ Y ) )
        = ( X4 = Y ) ) ) ).

% arccos_eq_iff
thf(fact_10013_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_10014_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_10015_arccos__less__arccos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_10016_arccos__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X4 ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X4 ) ) ) ) ).

% arccos_less_mono
thf(fact_10017_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_10018_arccos__cos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = X4 ) ) ) ).

% arccos_cos
thf(fact_10019_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_10020_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_10021_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_10022_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_10023_sin__arccos__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_10024_arccos__cos2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X4 )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = ( uminus_uminus_real @ X4 ) ) ) ) ).

% arccos_cos2
thf(fact_10025_arccos__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_minus
thf(fact_10026_arccos__def,axiom,
    ( arccos
    = ( ^ [Y4: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y4 ) ) ) ) ) ).

% arccos_def
thf(fact_10027_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_10028_arccos__minus__abs,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ).

% arccos_minus_abs
thf(fact_10029_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_10030_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_10031_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_10032_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_10033_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L3
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_10034_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_10035_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_10036_OR__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X4 @ Y ) ) ) ) ).

% OR_lower
thf(fact_10037_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_10038_int__or__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se1409905431419307370or_int @ I @ zero_zero_int )
      = I ) ).

% int_or_code(2)
thf(fact_10039_int__or__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ J )
      = J ) ).

% int_or_code(1)
thf(fact_10040_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_10041_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_10042_OR__upper,axiom,
    ! [X4: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( 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 @ X4 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_10043_or__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(4)
thf(fact_10044_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_10045_or__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(3)
thf(fact_10046_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_10047_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_10048_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_10049_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_10050_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( 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 @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_10051_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_10052_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_10053_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_10054_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_10055_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_10056_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_10057_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q7: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q7 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_10058_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_10059_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_10060_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N11: num] : ( some_num @ ( bit1 @ N11 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_10061_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_10062_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_10063_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_10064_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_10065_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_10066_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_10067_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_10068_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_10069_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_10070_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_10071_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
            @ ( bit_take_bit_num @ N4 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_10072_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_10073_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M5: num] :
          ( produc478579273971653890on_num
          @ ^ [A2: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P6: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
                      @ ( bit_take_bit_num @ O @ P6 ) )
                  @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
                  @ X )
              @ A2 )
          @ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_10074_rat__inverse__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_inverse_code
thf(fact_10075_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_10076_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu3: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_10077_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu3: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_10078_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_10079_quotient__of__denom__pos,axiom,
    ! [R3: rat,P5: int,Q5: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ord_less_int @ zero_zero_int @ Q5 ) ) ).

% quotient_of_denom_pos
thf(fact_10080_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N @ M4 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_10081_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_10082_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_10083_rat__uminus__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_uminus_code
thf(fact_10084_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P6: rat,Q7: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D5: int] : ( ord_less_int @ ( times_times_int @ A2 @ D5 ) @ ( times_times_int @ C5 @ B2 ) )
              @ ( quotient_of @ Q7 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_code
thf(fact_10085_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P6: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P6 ) ) ) ) ).

% rat_floor_code
thf(fact_10086_rat__abs__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int] : ( product_Pair_int_int @ ( abs_abs_int @ A2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_abs_code
thf(fact_10087_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P6: rat,Q7: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A2 @ D5 ) @ ( times_times_int @ C5 @ B2 ) )
              @ ( quotient_of @ Q7 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_eq_code
thf(fact_10088_rat__plus__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A2 @ D5 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_plus_code
thf(fact_10089_rat__minus__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A2 @ D5 ) @ ( times_times_int @ B2 @ C5 ) ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_minus_code
thf(fact_10090_normalize__denom__zero,axiom,
    ! [P5: int] :
      ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_10091_normalize__negative,axiom,
    ! [Q5: int,P5: int] :
      ( ( ord_less_int @ Q5 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q5 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q5 ) ) ) ) ) ).

% normalize_negative
thf(fact_10092_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P5: int,Q5: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ord_less_int @ zero_zero_int @ Q5 ) ) ).

% normalize_denom_pos
thf(fact_10093_normalize__crossproduct,axiom,
    ! [Q5: int,S2: int,P5: int,R3: int] :
      ( ( Q5 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q5 ) )
            = ( normalize @ ( product_Pair_int_int @ R3 @ S2 ) ) )
         => ( ( times_times_int @ P5 @ S2 )
            = ( times_times_int @ R3 @ Q5 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_10094_rat__times__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( times_times_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_times_code
thf(fact_10095_rat__divide__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ D5 ) @ ( times_times_int @ C5 @ B2 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_divide_code
thf(fact_10096_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M5: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_10097_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_10098_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_10099_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_10100_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_10101_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_10102_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_10103_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_10104_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_10105_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_10106_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_10107_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_10108_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_10109_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_10110_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_10111_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_10112_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_10113_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_10114_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_10115_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q5 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q5 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_10116_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N4: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_nat_def
thf(fact_10117_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_10118_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_10119_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_10120_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_10121_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_10122_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_10123_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_10124_int__sdiv__simps_I2_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% int_sdiv_simps(2)
thf(fact_10125_sdiv__int__0__div,axiom,
    ! [X4: int] :
      ( ( signed6714573509424544716de_int @ zero_zero_int @ X4 )
      = zero_zero_int ) ).

% sdiv_int_0_div
thf(fact_10126_sdiv__int__div__0,axiom,
    ! [X4: int] :
      ( ( signed6714573509424544716de_int @ X4 @ zero_zero_int )
      = zero_zero_int ) ).

% sdiv_int_div_0
thf(fact_10127_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_10128_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_10129_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( signed6714573509424544716de_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( signed6714573509424544716de_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_10130_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M9 @ M5 )
         => ! [N4: nat] :
              ( ( ord_less_eq_nat @ M9 @ N4 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_10131_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu4: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_10132_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_10133_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10134_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10135_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_10136_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_10137_min__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_10138_min__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_10139_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q5 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q5 ) @ ( times_times_nat @ N @ Q5 ) ) ) ).

% nat_mult_min_left
thf(fact_10140_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q5 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q5 ) ) ) ).

% nat_mult_min_right
thf(fact_10141_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10142_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_10143_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N @ M4 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10144_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10145_min__enat__simps_I2_J,axiom,
    ! [Q5: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q5 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_10146_min__enat__simps_I3_J,axiom,
    ! [Q5: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q5 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_10147_int__set__bits__K__False,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $false )
    = zero_zero_int ) ).

% int_set_bits_K_False
thf(fact_10148_int__set__bits__K__True,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $true )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_set_bits_K_True
thf(fact_10149_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_10150_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_10151_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_10152_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_10153_norm__assertion__simps_I4_J,axiom,
    ! [X4: assn] :
      ( ( inf_inf_assn @ X4 @ top_top_assn )
      = X4 ) ).

% norm_assertion_simps(4)
thf(fact_10154_norm__assertion__simps_I3_J,axiom,
    ! [X4: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X4 )
      = X4 ) ).

% norm_assertion_simps(3)
thf(fact_10155_ent__star__mono__true,axiom,
    ! [A3: assn,A8: assn,B4: assn,B10: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ A8 @ top_top_assn ) )
     => ( ( entails @ B4 @ ( times_times_assn @ B10 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A3 @ B4 ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A8 @ B10 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_10156_ent__refl__true,axiom,
    ! [A3: assn] : ( entails @ A3 @ ( times_times_assn @ A3 @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_10157_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entails @ ( times_times_assn @ P @ R ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(1)
thf(fact_10158_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(2)
thf(fact_10159_merge__true__star__ctx,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P ) )
      = ( times_times_assn @ top_top_assn @ P ) ) ).

% merge_true_star_ctx
thf(fact_10160_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_10161_norm__assertion__simps_I12_J,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ X4 @ top_top_assn )
      = top_top_assn ) ).

% norm_assertion_simps(12)
thf(fact_10162_norm__assertion__simps_I11_J,axiom,
    ! [X4: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X4 )
      = top_top_assn ) ).

% norm_assertion_simps(11)
thf(fact_10163_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_10164_surj__list__encode,axiom,
    ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
    = top_top_set_nat ) ).

% surj_list_encode
thf(fact_10165_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_10166_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_10167_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_10168_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_10169_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_10170_suminf__eq__SUP__real,axiom,
    ! [X7: nat > real] :
      ( ( summable_real @ X7 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X7 @ I2 ) )
       => ( ( suminf_real @ X7 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X7 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_10171_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_10172_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_10173_ones,axiom,
    ! [N: nat,F: nat > $o] :
      ( ! [N6: nat] :
          ( ( ord_less_eq_nat @ N @ N6 )
         => ( F @ N6 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_10174_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N2: nat] :
            ~ ! [N5: nat] :
                ( ( ord_less_eq_nat @ N2 @ N5 )
               => ~ ( F @ N5 ) )
       => ~ ! [N2: nat] :
              ~ ! [N5: nat] :
                  ( ( ord_less_eq_nat @ N2 @ N5 )
                 => ( F @ N5 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_10175_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
          ( ? [N4: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N4 @ N11 )
             => ~ ( F4 @ N11 ) )
          | ? [N4: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N4 @ N11 )
             => ( F4 @ N11 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_10176_zeros,axiom,
    ! [N: nat,F: nat > $o] :
      ( ! [N6: nat] :
          ( ( ord_less_eq_nat @ N @ N6 )
         => ~ ( F @ N6 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_10177_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
        ? [N4: nat] :
          ( ! [N11: nat] :
              ( ( ord_less_eq_nat @ N4 @ N11 )
             => ~ ( F4 @ N11 ) )
          | ! [N11: nat] :
              ( ( ord_less_eq_nat @ N4 @ N11 )
             => ( F4 @ N11 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_10178_wf__set__bits__int__const,axiom,
    ! [B: $o] :
      ( bit_wf_set_bits_int
      @ ^ [Uu3: nat] : B ) ).

% wf_set_bits_int_const
thf(fact_10179_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_10180_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_10181_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_10182_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_10183_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_10184_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_10185_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_10186_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_10187_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_10188_binomial__def,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_10189_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_10190_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M8 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_10191_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M8 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_10192_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_10193_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
     => ( A3
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_10194_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_10195_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_10196_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_10197_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_10198_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_10199_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L3: list_nat] :
                ( ( ( size_size_list_nat @ L3 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L3 )
                  = N7 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L3 )
                    = N7 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
                    = N7 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10200_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L3: list_nat] :
              ( ( ( size_size_list_nat @ L3 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L3 )
                = N7 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).

% card_length_sum_list
thf(fact_10201_card__num0,axiom,
    ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
    = zero_zero_nat ) ).

% card_num0
thf(fact_10202_card__nat,axiom,
    ( ( finite_card_nat @ top_top_set_nat )
    = zero_zero_nat ) ).

% card_nat
thf(fact_10203_card__literal,axiom,
    ( ( finite_card_literal @ top_top_set_literal )
    = zero_zero_nat ) ).

% card_literal
thf(fact_10204_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).

% Inf_nat_def1
thf(fact_10205_inj__on__set__encode,axiom,
    inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).

% inj_on_set_encode

% Helper facts (71)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y: int] :
      ( ( if_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y: int] :
      ( ( if_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y: nat] :
      ( ( if_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y: nat] :
      ( ( if_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y: num] :
      ( ( if_num @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y: num] :
      ( ( if_num @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y: rat] :
      ( ( if_rat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y: rat] :
      ( ( if_rat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y: real] :
      ( ( if_real @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y: real] :
      ( ( if_real @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X4: set_o,Y: set_o] :
      ( ( if_set_o @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X4: set_o,Y: set_o] :
      ( ( if_set_o @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X4: assn,Y: assn] :
      ( ( if_assn @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X4: assn,Y: assn] :
      ( ( if_assn @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y: complex] :
      ( ( if_complex @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y: complex] :
      ( ( if_complex @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X4: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X4: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_I_Eo_J_T,axiom,
    ! [X4: option_o,Y: option_o] :
      ( ( if_option_o @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_I_Eo_J_T,axiom,
    ! [X4: option_o,Y: option_o] :
      ( ( if_option_o @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X4: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X4: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X4: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Real__Oreal_J_T,axiom,
    ! [X4: set_real,Y: set_real] :
      ( ( if_set_real @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Real__Oreal_J_T,axiom,
    ! [X4: set_real,Y: set_real] :
      ( ( if_set_real @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Int__Oint_J_T,axiom,
    ! [X4: option_int,Y: option_int] :
      ( ( if_option_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Int__Oint_J_T,axiom,
    ! [X4: option_int,Y: option_int] :
      ( ( if_option_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X4: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X4: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Real__Oreal_J_T,axiom,
    ! [X4: option_real,Y: option_real] :
      ( ( if_option_real @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Real__Oreal_J_T,axiom,
    ! [X4: option_real,Y: option_real] :
      ( ( if_option_real @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Complex__Ocomplex_J_T,axiom,
    ! [X4: set_complex,Y: set_complex] :
      ( ( if_set_complex @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Complex__Ocomplex_J_T,axiom,
    ! [X4: set_complex,Y: set_complex] :
      ( ( if_set_complex @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X4: heap_Time_Heap_o,Y: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X4: heap_Time_Heap_o,Y: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: set_Code_integer,Y: set_Code_integer] :
      ( ( if_set_Code_integer @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: set_Code_integer,Y: set_Code_integer] :
      ( ( if_set_Code_integer @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X4: set_VEBT_VEBT,Y: set_VEBT_VEBT] :
      ( ( if_set_VEBT_VEBT @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X4: set_VEBT_VEBT,Y: set_VEBT_VEBT] :
      ( ( if_set_VEBT_VEBT @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X4: heap_Time_Heap_nat,Y: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X4: heap_Time_Heap_nat,Y: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X4: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X4: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X4: option_VEBT_VEBT,Y: option_VEBT_VEBT] :
      ( ( if_option_VEBT_VEBT @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__VEBT____Definitions__OVEBT_J_T,axiom,
    ! [X4: option_VEBT_VEBT,Y: option_VEBT_VEBT] :
      ( ( if_option_VEBT_VEBT @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X4: option_VEBT_VEBTi,Y: option_VEBT_VEBTi] :
      ( ( if_option_VEBT_VEBTi @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X4: option_VEBT_VEBTi,Y: option_VEBT_VEBTi] :
      ( ( if_option_VEBT_VEBTi @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X4: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X4: heap_T2636463487746394924on_nat,Y: heap_T2636463487746394924on_nat] :
      ( ( if_Hea5867803462524415986on_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X4: heap_T2636463487746394924on_nat,Y: heap_T2636463487746394924on_nat] :
      ( ( if_Hea5867803462524415986on_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( if_opt6109864365331422477at_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( if_opt6109864365331422477at_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X4 @ Y )
      = X4 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ v ) @ ( suc @ zero_zero_nat ) @ vg @ vh ) @ tia ) @ ( vEBT_VEBT_vebt_succi @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ v ) @ ( suc @ zero_zero_nat ) @ vg @ vh ) @ tia @ vi )
    @ ^ [R2: option_nat] :
        ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ v ) @ ( suc @ zero_zero_nat ) @ vg @ vh ) @ tia )
        @ ( pure_assn
          @ ( R2
            = ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ v ) @ ( suc @ zero_zero_nat ) @ vg @ vh ) @ vi ) ) ) ) ) ).

%------------------------------------------------------------------------------