TPTP Problem File: ITP219^3.p

%------------------------------------------------------------------------------
% File     : ITP219^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_List_Assn 00252_007651
% 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    : 0062_VEBT_List_Assn_00252_007651 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.2.0, 0.31 v8.1.0
% Syntax   : Number of formulae    : 7184 (2817 unt;1397 typ;   0 def)
%            Number of atoms       : 16077 (6632 equ;   2 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 55945 (1791   ~; 316   |;1080   &;46232   @)
%                                         (   0 <=>;6526  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   6 avg)
%            Number of types       :  237 ( 236 usr)
%            Number of type conns  : 4116 (4116   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1165 (1161 usr;  89 con; 0-6 aty)
%            Number of variables   : 16777 ( 981   ^;15414   !; 382   ?;16777   :)
% 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
%            2022-02-17 17:07:01.933
%------------------------------------------------------------------------------
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    relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).

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

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

thf(sy_c_Assertions_Owand__assn,type,
    wand_assn: assn > assn > assn ).

thf(sy_c_Assertions_Owand__raw,type,
    wand_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

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

thf(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > assn > assn > $o ).

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

thf(sy_c_Automation_OSLN,type,
    sln: assn ).

thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
    bNF_Ca8331644756375544342r_on_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

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

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_Oconcat__bit,type,
    bit_concat_bit: nat > int > 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_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

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__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

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

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

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

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

thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: code_natural > nat ).

thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: nat > code_natural ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

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

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

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_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

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

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

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

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

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

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

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

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
    cofinite_nat: filter_nat ).

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

thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

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

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    filter3212408913953519116et_nat: ( nat > set_nat ) > filter_set_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofinite__subsets__at__top_001t__Nat__Onat,type,
    finite3254316476582989077op_nat: set_nat > filter_set_nat ).

thf(sy_c_Filter_Ofrequently_001t__Nat__Onat,type,
    frequently_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

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

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

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

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Int__Oint,type,
    comp_o_o_int: ( $o > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Nat__Onat,type,
    comp_o_o_nat: ( $o > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
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    inf_in7913087082777306421at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    sup_sup_filter_nat: filter_nat > filter_nat > filter_nat ).

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__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su5209123915105501825_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).

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__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__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__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    sup_su5769328420594410459od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    sup_su1735938885662425829n_assn: set_Pr5949110396991348497n_assn > set_Pr5949110396991348497n_assn > set_Pr5949110396991348497n_assn ).

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__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    sup_su6664651903789602459at_nat: set_Pr5564308138774400199at_nat > set_Pr5564308138774400199at_nat > set_Pr5564308138774400199at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su3642409539654194069at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_List_Oall__interval__int,type,
    all_interval_int: ( int > $o ) > int > int > $o ).

thf(sy_c_List_Oall__interval__nat,type,
    all_interval_nat: ( nat > $o ) > nat > nat > $o ).

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

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

thf(sy_c_List_Obutlast_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    butlas8287983287669047583_nat_o: list_P7985473006766602707_nat_o > list_P7985473006766602707_nat_o ).

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

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

thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    butlas3012047794866324995n_assn: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    butlas5569151904373679443at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    butlas2561766218336428473at_nat: list_P5464809261938338413at_nat > list_P5464809261938338413at_nat ).

thf(sy_c_List_Obutlast_001tf__c,type,
    butlast_c: list_c > list_c ).

thf(sy_c_List_Obutlast_001tf__d,type,
    butlast_d: list_d > list_d ).

thf(sy_c_List_Ocoset_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    coset_9017067850271954240_nat_o: list_P7985473006766602707_nat_o > set_Pr4532377907799695533_nat_o ).

thf(sy_c_List_Ocoset_001_Eo,type,
    coset_o: list_o > set_o ).

thf(sy_c_List_Ocoset_001t__Int__Oint,type,
    coset_int: list_int > set_int ).

thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
    coset_nat: list_nat > set_nat ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    coset_703488708621576356n_assn: list_P8527749157015355191n_assn > set_Pr5949110396991348497n_assn ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    coset_6809516720700967924at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    coset_5307619257043544794at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_List_Ocount__list_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    count_5695430372156137289_nat_o: list_P7985473006766602707_nat_o > ( produc3658429121746597890et_nat > $o ) > nat ).

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

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

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    count_2530312006313534765n_assn: list_P8527749157015355191n_assn > produc6575502325842934193n_assn > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    count_1483299019173949923at_nat: list_P5464809261938338413at_nat > produc3843707927480180839at_nat > nat ).

thf(sy_c_List_Odistinct_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    distin299582740263731014_nat_o: list_P7985473006766602707_nat_o > $o ).

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

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

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

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

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

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    distin4809818325527169450n_assn: list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    distin3744728255968310194nt_int: list_P5707943133018811711nt_int > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    distin7922579275477506902nt_nat: list_P8198026277950538467nt_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    distin2745374544067255638at_int: list_P3521021558325789923at_int > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).

thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    distin5355857245021852384at_nat: list_P5464809261938338413at_nat > $o ).

thf(sy_c_List_Odistinct_001tf__c,type,
    distinct_c: list_c > $o ).

thf(sy_c_List_Odistinct_001tf__d,type,
    distinct_d: list_d > $o ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

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

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

thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    enumer9200277086812776726n_assn: nat > list_P8527749157015355191n_assn > list_P7711101519850957926n_assn ).

thf(sy_c_List_Oenumerate_001tf__c,type,
    enumerate_c: nat > list_c > list_P2993859830824179253_nat_c ).

thf(sy_c_List_Oenumerate_001tf__d,type,
    enumerate_d: nat > list_d > list_P3064893870870525238_nat_d ).

thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat2: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
    fold_int_int: ( int > int > int ) > list_int > int > int ).

thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

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

thf(sy_c_List_Ofoldl_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    foldl_1542389171486570077nteger: ( code_integer > code_integer > code_integer ) > code_integer > list_Code_integer > code_integer ).

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

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

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

thf(sy_c_List_Ofoldl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    foldl_set_o_set_o: ( set_o > set_o > set_o ) > set_o > list_set_o > set_o ).

thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    foldl_6819690284573351271et_int: ( set_int > set_int > set_int ) > set_int > list_set_int > set_int ).

thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    foldl_4988731653086973103et_nat: ( set_nat > set_nat > set_nat ) > set_nat > list_set_nat > set_nat ).

thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    foldl_5307762784932486881at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    foldl_4541198706405875041at_nat: ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat > list_s4242580248156217677at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_List_Ofoldr_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    foldr_assn_assn: ( assn > assn > assn ) > list_assn > assn > assn ).

thf(sy_c_List_Oinsert_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    insert8021100735160035049_nat_o: ( produc3658429121746597890et_nat > $o ) > list_P7985473006766602707_nat_o > list_P7985473006766602707_nat_o ).

thf(sy_c_List_Oinsert_001_Eo,type,
    insert_o: $o > list_o > list_o ).

thf(sy_c_List_Oinsert_001t__Int__Oint,type,
    insert_int: int > list_int > list_int ).

thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    insert3246601298802261197n_assn: produc6575502325842934193n_assn > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8944034826898310173at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    insert4149901806602606979at_nat: produc3843707927480180839at_nat > list_P5464809261938338413at_nat > list_P5464809261938338413at_nat ).

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

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

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

thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    lenlex6671093235567761724n_assn: set_Pr8116311503512020807n_assn > set_Pr6863447823992824007n_assn ).

thf(sy_c_List_Olenlex_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lenlex1357538814655152620at_nat: set_Pr4329608150637261639at_nat > set_Pr4333006031979791559at_nat ).

thf(sy_c_List_Olenlex_001tf__c,type,
    lenlex_c: set_Product_prod_c_c > set_Pr2058341694840537735list_c ).

thf(sy_c_List_Olenlex_001tf__d,type,
    lenlex_d: set_Product_prod_d_d > set_Pr5674772971416283335list_d ).

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

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

thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    lex_Pr7559194461235888923n_assn: set_Pr8116311503512020807n_assn > set_Pr6863447823992824007n_assn ).

thf(sy_c_List_Olex_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lex_se2245640040323279819at_nat: set_Pr4329608150637261639at_nat > set_Pr4333006031979791559at_nat ).

thf(sy_c_List_Olex_001tf__c,type,
    lex_c: set_Product_prod_c_c > set_Pr2058341694840537735list_c ).

thf(sy_c_List_Olex_001tf__d,type,
    lex_d: set_Product_prod_d_d > set_Pr5674772971416283335list_d ).

thf(sy_c_List_Olexn_001t__Int__Oint,type,
    lexn_int: set_Pr958786334691620121nt_int > nat > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olexn_001t__Nat__Onat,type,
    lexn_nat: set_Pr1261947904930325089at_nat > nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexn_001tf__c,type,
    lexn_c: set_Product_prod_c_c > nat > set_Pr2058341694840537735list_c ).

thf(sy_c_List_Olexn_001tf__d,type,
    lexn_d: set_Product_prod_d_d > nat > set_Pr5674772971416283335list_d ).

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_Nat__Bijection_Oset__encode,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_I_Eo_J,type,
    ord_min_set_o: set_o > set_o > set_o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Int__Oint_J,type,
    ord_min_set_int: set_int > set_int > set_int ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_min_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_mi4194197829855655752at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    ord_mi6693596345299712430at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

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__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_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Onatural_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc2252593628808123835atural: ( code_natural > option6357759511663192854e_term ) > produc7822875418678951345atural > produc8731074985263844745atural ).

thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
    produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_M_062_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    produc6115016301897879072n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > produc4530688714081574576n_assn ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    produc7368190662567826135_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc4928098042776334183_nat_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc2245416461498447860et_nat: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc5001842942810119800et_nat: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat ).

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

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

thf(sy_c_Product__Type_OPair_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    produc118845697133431529n_assn: assn > assn > produc6575502325842934193n_assn ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    produc584006145561248582it_nat: heap_e7401611519738050253t_unit > nat > produc6653097349344004940it_nat ).

thf(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: heap_e7401611519738050253t_unit > set_nat > produc3658429121746597890et_nat ).

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

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

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    produc3313821056773121708n_assn: int > produc6575502325842934193n_assn > produc3167338625816234618n_assn ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc3525228511222278652at_nat: int > product_prod_nat_nat > produc1062882101152166346at_nat ).

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__Int__Oint_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4542114716404682293st_nat: list_int > list_nat > produc3676724955757786621st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    produc4510294837168919970n_assn: list_int > list_P8527749157015355191n_assn > produc8582807064127716720n_assn ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Int__Oint_J,type,
    produc7739558402351520821st_int: list_nat > list_int > produc8561936516282095101st_int ).

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

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    produc7706768604797473534n_assn: list_nat > list_P8527749157015355191n_assn > produc2555908794901494476n_assn ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_Itf__c_J,type,
    produc7723716018658481613list_c: list_nat > list_c > produc1655478831073744795list_c ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_Itf__d_J,type,
    produc7723716022961710414list_d: list_nat > list_d > produc1726512871120090780list_d ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_001t__List__Olist_It__Int__Oint_J,type,
    produc1996262876970182428st_int: list_P8527749157015355191n_assn > list_int > produc4164172529858598818st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc6174113896479379136st_nat: list_P8527749157015355191n_assn > list_nat > produc6654255674790325574st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    produc3014731801419794263n_assn: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > produc337509097454979815n_assn ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7536900900485677911at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat > produc424102278133772007at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__c_J_001t__List__Olist_It__Int__Oint_J,type,
    produc1948897715771513579st_int: list_c > list_int > produc8499486975545879281st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__c_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc6126748735280710287st_nat: list_c > list_nat > produc1766198083622830229st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__c_J_001t__List__Olist_Itf__c_J,type,
    produc5515445818230282391list_c: list_c > list_c > produc7985223085803476903list_c ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__c_J_001t__List__Olist_Itf__d_J,type,
    produc5515445822533511192list_d: list_c > list_d > produc8056257125849822888list_d ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__d_J_001t__List__Olist_It__Int__Oint_J,type,
    produc2615797191311421932st_int: list_d > list_int > produc9166386451085787634st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__d_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc6793648210820618640st_nat: list_d > list_nat > produc2433097559162738582st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__d_J_001t__List__Olist_Itf__c_J,type,
    produc242965416994095062list_c: list_d > list_c > produc2712742684567289574list_c ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__d_J_001t__List__Olist_Itf__d_J,type,
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thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__c,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__d,type,
    product_Pair_nat_d: nat > d > product_prod_nat_d ).

thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Random_Olog__rel,type,
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thf(sy_c_Random_Ominus__shift,type,
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thf(sy_c_Random_Onext,type,
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thf(sy_c_Random_Orange,type,
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thf(sy_c_Random_Osplit__seed,type,
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thf(sy_c_Rat_OAbs__Rat,type,
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thf(sy_c_Rat_OFrct,type,
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thf(sy_c_Rat_ORep__Rat,type,
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thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oof__int,type,
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thf(sy_c_Rat_Opcr__rat,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Rat_Oratrel,type,
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thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
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thf(sy_c_Relation_OField_001t__Int__Oint,type,
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% Relevant facts (5751)
thf(fact_0_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H )
         => ( rep_assn @ Q @ H ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_1__092_060open_062P_A_092_060Longrightarrow_062_092_060_094sub_062A_Afalse_092_060close_062,axiom,
    entails @ p @ bot_bot_assn ).

% \<open>P \<Longrightarrow>\<^sub>A false\<close>
thf(fact_2_assms_I3_J,axiom,
    member_nat @ i2 @ i ).

% assms(3)
thf(fact_3_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( ( entails @ A @ B )
     => ( ( entails @ B @ A )
       => ( A = B ) ) ) ).

% ent_iffI
thf(fact_4_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_5_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_6_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_7_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_8_ent__fwd,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H2 )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% ent_fwd
thf(fact_9_entailsD,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H2 )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% entailsD
thf(fact_10_entails__def,axiom,
    ( entails
    = ( ^ [P2: assn,Q2: assn] :
        ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H3 )
         => ( rep_assn @ Q2 @ H3 ) ) ) ) ).

% entails_def
thf(fact_11_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H3: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H3 ) ) ) ).

% ent_false_iff
thf(fact_12_assms_I4_J,axiom,
    entails @ ( times_times_assn @ ( vEBT_L824818263033719201sn_c_d @ i @ a @ ( list_update_c @ xs @ i2 @ x ) @ ( list_update_d @ xsi @ i2 @ xi ) ) @ f ) @ q ).

% assms(4)
thf(fact_13_ent__imp__entt,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entailst @ P @ Q ) ) ).

% ent_imp_entt
thf(fact_14_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_15_entt__refl,axiom,
    ! [A: assn] : ( entailst @ A @ A ) ).

% entt_refl
thf(fact_16_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_17_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_18_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A2 @ C ) @ B2 ) ) ).

% assn_aci(10)
thf(fact_19_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C ) )
      = ( times_times_assn @ B2 @ ( times_times_assn @ A2 @ C ) ) ) ).

% star_aci(3)
thf(fact_20_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A3: assn,B3: assn] : ( times_times_assn @ B3 @ A3 ) ) ) ).

% star_aci(2)
thf(fact_21_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H2 ) ).

% mod_false
thf(fact_22_mod__starD,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A @ H1 )
          & ( rep_assn @ B @ H22 ) ) ) ).

% mod_starD
thf(fact_23_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B2 ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B2 @ H_2 ) ) ) ).

% mod_starE
thf(fact_24_entt__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entailst @ A @ B )
     => ( ( entailst @ B @ C2 )
       => ( entailst @ A @ C2 ) ) ) ).

% entt_trans
thf(fact_25_star__assoc,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C ) ) ) ).

% star_assoc
thf(fact_26_entt__fr__drop,axiom,
    ! [F: assn,F2: assn,A: assn] :
      ( ( entailst @ F @ F2 )
     => ( entailst @ ( times_times_assn @ F @ A ) @ F2 ) ) ).

% entt_fr_drop
thf(fact_27_entt__fr__refl,axiom,
    ! [F: assn,F2: assn,A: assn] :
      ( ( entailst @ F @ F2 )
     => ( entailst @ ( times_times_assn @ F @ A ) @ ( times_times_assn @ F2 @ A ) ) ) ).

% entt_fr_refl
thf(fact_28_entt__frame__fwd,axiom,
    ! [P: assn,Q: assn,A: assn,F: assn,B: assn] :
      ( ( entailst @ P @ Q )
     => ( ( entailst @ A @ ( times_times_assn @ P @ F ) )
       => ( ( entailst @ ( times_times_assn @ Q @ F ) @ B )
         => ( entailst @ A @ B ) ) ) ) ).

% entt_frame_fwd
thf(fact_29_entt__star__mono,axiom,
    ! [A: assn,B: assn,C2: assn,D: assn] :
      ( ( entailst @ A @ B )
     => ( ( entailst @ C2 @ D )
       => ( entailst @ ( times_times_assn @ A @ C2 ) @ ( times_times_assn @ B @ D ) ) ) ) ).

% entt_star_mono
thf(fact_30_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P2: assn,Q2: assn] : ( times_times_assn @ Q2 @ P2 ) ) ) ).

% assn_times_comm
thf(fact_31_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_32_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P: assn,R: assn,F: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P @ R )
       => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
         => ( rep_assn @ ( times_times_assn @ R @ F ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_33_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F: assn,Q: assn] :
      ( ( entails @ P @ R )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( ( entails @ ( times_times_assn @ R @ F ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_34_ent__star__mono,axiom,
    ! [P: assn,P3: assn,Q: assn,Q3: assn] :
      ( ( entails @ P @ P3 )
     => ( ( entails @ Q @ Q3 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P3 @ Q3 ) ) ) ) ).

% ent_star_mono
thf(fact_35_fr__rot__rhs,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entails @ A @ ( times_times_assn @ B @ C2 ) )
     => ( entails @ A @ ( times_times_assn @ C2 @ B ) ) ) ).

% fr_rot_rhs
thf(fact_36_fr__refl,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entails @ A @ B )
     => ( entails @ ( times_times_assn @ A @ C2 ) @ ( times_times_assn @ B @ C2 ) ) ) ).

% fr_refl
thf(fact_37_fr__rot,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entails @ ( times_times_assn @ A @ B ) @ C2 )
     => ( entails @ ( times_times_assn @ B @ A ) @ C2 ) ) ).

% fr_rot
thf(fact_38_assms_I1_J,axiom,
    entails @ p @ ( times_times_assn @ ( times_times_assn @ ( a @ x @ xi ) @ ( vEBT_L824818263033719201sn_c_d @ ( minus_minus_set_nat @ i @ ( insert_nat2 @ i2 @ bot_bot_set_nat ) ) @ a @ xs @ xsi ) ) @ f ) ).

% assms(1)
thf(fact_39_list__update__overwrite,axiom,
    ! [Xs: list_c,I: nat,X: c,Y: c] :
      ( ( list_update_c @ ( list_update_c @ Xs @ I @ X ) @ I @ Y )
      = ( list_update_c @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_40_list__update__overwrite,axiom,
    ! [Xs: list_d,I: nat,X: d,Y: d] :
      ( ( list_update_d @ ( list_update_d @ Xs @ I @ X ) @ I @ Y )
      = ( list_update_d @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_41_False,axiom,
    ( ( size_size_list_c @ xs )
   != ( size_size_list_d @ xsi ) ) ).

% False
thf(fact_42_assms_I2_J,axiom,
    ord_less_nat @ i2 @ ( size_size_list_c @ xs ) ).

% assms(2)
thf(fact_43_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( ( fI_QUERY @ P @ Q @ F )
     => ( entails @ P @ ( times_times_assn @ Q @ F ) ) ) ).

% frame_inference_init
thf(fact_44_mem__Collect__eq,axiom,
    ! [A2: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A2 @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( collec939566748876313656_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_54_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_55_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_56_Collect__cong,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec939566748876313656_nat_o @ P )
        = ( collec939566748876313656_nat_o @ Q ) ) ) ).

% Collect_cong
thf(fact_57_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P2: assn,Q2: assn,F3: assn] : ( entails @ P2 @ ( times_times_assn @ Q2 @ F3 ) ) ) ) ).

% FI_QUERY_def
thf(fact_58_ent__wand__frameI,axiom,
    ! [Q: assn,R: assn,F: assn,S: assn,P: assn,X4: assn] :
      ( ( entails @ ( times_times_assn @ ( wand_assn @ Q @ R ) @ F ) @ S )
     => ( ( entails @ P @ ( times_times_assn @ F @ X4 ) )
       => ( ( entails @ ( times_times_assn @ Q @ X4 ) @ R )
         => ( entails @ P @ S ) ) ) ) ).

% ent_wand_frameI
thf(fact_59_ent__wandI,axiom,
    ! [Q: assn,P: assn,R: assn] :
      ( ( entails @ ( times_times_assn @ Q @ P ) @ R )
     => ( entails @ P @ ( wand_assn @ Q @ R ) ) ) ).

% ent_wandI
thf(fact_60_ent__mp,axiom,
    ! [P: assn,Q: assn] : ( entails @ ( times_times_assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).

% ent_mp
thf(fact_61_length__list__update,axiom,
    ! [Xs: list_c,I: nat,X: c] :
      ( ( size_size_list_c @ ( list_update_c @ Xs @ I @ X ) )
      = ( size_size_list_c @ Xs ) ) ).

% length_list_update
thf(fact_62_length__list__update,axiom,
    ! [Xs: list_d,I: nat,X: d] :
      ( ( size_size_list_d @ ( list_update_d @ Xs @ I @ X ) )
      = ( size_size_list_d @ Xs ) ) ).

% length_list_update
thf(fact_63_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_64_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_65_order__less__imp__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_66_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_67_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_68_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_69_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_70_order__less__imp__not__eq2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_71_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_72_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_73_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_74_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_75_order__less__imp__not__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_76_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_77_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_78_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_79_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_80_linorder__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
      | ( X = Y )
      | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_81_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_82_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_83_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_84_order__less__imp__triv,axiom,
    ! [X: code_integer,Y: code_integer,P: $o] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_85_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P: $o] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_86_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_87_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_88_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_89_order__less__not__sym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_90_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_91_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_92_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_93_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_94_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_95_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > assn,C: assn] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_96_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_less_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_97_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_98_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_less_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_99_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > code_integer,C: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_100_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > assn,C: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_101_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > num,C: num] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_102_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > nat,C: nat] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_103_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > int,C: int] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_104_order__less__subst1,axiom,
    ! [A2: code_integer,F4: code_integer > code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_105_order__less__subst1,axiom,
    ! [A2: code_integer,F4: assn > code_integer,B2: assn,C: assn] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_106_order__less__subst1,axiom,
    ! [A2: code_integer,F4: num > code_integer,B2: num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_107_order__less__subst1,axiom,
    ! [A2: code_integer,F4: nat > code_integer,B2: nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_108_order__less__subst1,axiom,
    ! [A2: code_integer,F4: int > code_integer,B2: int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_109_order__less__subst1,axiom,
    ! [A2: assn,F4: code_integer > assn,B2: code_integer,C: code_integer] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_110_order__less__subst1,axiom,
    ! [A2: assn,F4: assn > assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_111_order__less__subst1,axiom,
    ! [A2: assn,F4: num > assn,B2: num,C: num] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_112_order__less__subst1,axiom,
    ! [A2: assn,F4: nat > assn,B2: nat,C: nat] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_113_order__less__subst1,axiom,
    ! [A2: assn,F4: int > assn,B2: int,C: int] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_114_order__less__irrefl,axiom,
    ! [X: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ X @ X ) ).

% order_less_irrefl
thf(fact_115_order__less__irrefl,axiom,
    ! [X: assn] :
      ~ ( ord_less_assn @ X @ X ) ).

% order_less_irrefl
thf(fact_116_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_117_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_118_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_119_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_120_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > assn,C: assn] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_121_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_122_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_123_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > int,C: int] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_124_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > code_integer,C: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_125_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > assn,C: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_126_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > num,C: num] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_127_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > nat,C: nat] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_128_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > int,C: int] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_129_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F4: code_integer > code_integer,B2: code_integer,C: code_integer] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_130_ord__eq__less__subst,axiom,
    ! [A2: assn,F4: code_integer > assn,B2: code_integer,C: code_integer] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_131_ord__eq__less__subst,axiom,
    ! [A2: num,F4: code_integer > num,B2: code_integer,C: code_integer] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_132_ord__eq__less__subst,axiom,
    ! [A2: nat,F4: code_integer > nat,B2: code_integer,C: code_integer] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_133_ord__eq__less__subst,axiom,
    ! [A2: int,F4: code_integer > int,B2: code_integer,C: code_integer] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_134_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F4: assn > code_integer,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_135_ord__eq__less__subst,axiom,
    ! [A2: assn,F4: assn > assn,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_136_ord__eq__less__subst,axiom,
    ! [A2: num,F4: assn > num,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_137_ord__eq__less__subst,axiom,
    ! [A2: nat,F4: assn > nat,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_138_ord__eq__less__subst,axiom,
    ! [A2: int,F4: assn > int,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_139_order__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_140_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_141_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_142_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_143_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_144_order__less__asym_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ~ ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_145_order__less__asym_H,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ~ ( ord_less_assn @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_146_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_147_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_148_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_149_linorder__neq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      = ( ( ord_le6747313008572928689nteger @ X @ Y )
        | ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_150_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_151_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_152_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_153_order__less__asym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_asym
thf(fact_154_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_asym
thf(fact_155_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_156_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_157_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_158_linorder__neqE,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_159_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_160_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_161_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_162_neq__if__length__neq,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( ( size_size_list_c @ Xs )
       != ( size_size_list_c @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_163_neq__if__length__neq,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( ( size_size_list_d @ Xs )
       != ( size_size_list_d @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_164_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_165_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_166_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_c] :
      ( ( size_size_list_c @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_167_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_d] :
      ( ( size_size_list_d @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_168_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_169_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] :
      ( ( size_size_list_int @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_170_length__induct,axiom,
    ! [P: list_c > $o,Xs: list_c] :
      ( ! [Xs2: list_c] :
          ( ! [Ys2: list_c] :
              ( ( ord_less_nat @ ( size_size_list_c @ Ys2 ) @ ( size_size_list_c @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_171_length__induct,axiom,
    ! [P: list_d > $o,Xs: list_d] :
      ( ! [Xs2: list_d] :
          ( ! [Ys2: list_d] :
              ( ( ord_less_nat @ ( size_size_list_d @ Ys2 ) @ ( size_size_list_d @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_172_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_173_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_174_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_175_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_177_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_178_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_179_order_Ostrict__implies__not__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_180_order_Ostrict__implies__not__eq,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_181_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_182_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_183_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_184_dual__order_Ostrict__trans,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C @ B2 )
       => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_185_dual__order_Ostrict__trans,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( ord_less_assn @ C @ B2 )
       => ( ord_less_assn @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_186_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_187_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_188_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_189_not__less__iff__gr__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_190_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_191_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_192_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_193_order_Ostrict__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_194_order_Ostrict__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ord_less_assn @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_195_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_196_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_197_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_198_linorder__less__wlog,axiom,
    ! [P: code_integer > code_integer > $o,A2: code_integer,B2: code_integer] :
      ( ! [A4: code_integer,B4: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: code_integer] : ( P @ A4 @ A4 )
       => ( ! [A4: code_integer,B4: code_integer] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_199_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A2: num,B2: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num] : ( P @ A4 @ A4 )
       => ( ! [A4: num,B4: num] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_200_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_201_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B4: int] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_202_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P2: nat > $o] :
        ? [N2: nat] :
          ( ( P2 @ N2 )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
             => ~ ( P2 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_203_dual__order_Oirrefl,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_204_dual__order_Oirrefl,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_205_dual__order_Oirrefl,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_206_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_207_dual__order_Oirrefl,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_208_dual__order_Oasym,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_209_dual__order_Oasym,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ~ ( ord_less_assn @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_210_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ~ ( ord_less_num @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_211_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_212_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ~ ( ord_less_int @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_213_linorder__cases,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( X != Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_214_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_215_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_216_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_217_antisym__conv3,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ Y @ X )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_218_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_219_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_220_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_221_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( ( ord_less_nat @ Y3 @ X3 )
             => ( P @ Y3 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_222_ord__less__eq__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_223_ord__less__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_assn @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_224_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_225_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_226_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_227_ord__eq__less__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 = B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_228_ord__eq__less__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( A2 = B2 )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ord_less_assn @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_229_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_230_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_231_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_232_order_Oasym,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ~ ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% order.asym
thf(fact_233_order_Oasym,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ~ ( ord_less_assn @ B2 @ A2 ) ) ).

% order.asym
thf(fact_234_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order.asym
thf(fact_235_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_236_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order.asym
thf(fact_237_less__imp__neq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_238_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_239_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_240_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_241_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_242_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_243_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_244_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).

% lt_ex
thf(fact_245_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( A2 != bot_bo228742789529271731at_nat )
      = ( ord_le2604355607129572851at_nat @ bot_bo228742789529271731at_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_246_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_247_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_248_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_249_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_250_bot_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_251_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_252_bot_Oextremum__strict,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A2 @ bot_bo228742789529271731at_nat ) ).

% bot.extremum_strict
thf(fact_253_bot_Oextremum__strict,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_254_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_255_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_256_bot_Oextremum__strict,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_257_bot_Oextremum__strict,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ A2 @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_258_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_259_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_c,A: c > c > assn] :
      ( ( ( size_size_list_c @ L )
       != ( size_size_list_c @ Li ) )
     => ( ( vEBT_L7968352277243653187sn_c_c @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_260_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_d,A: c > d > assn] :
      ( ( ( size_size_list_c @ L )
       != ( size_size_list_d @ Li ) )
     => ( ( vEBT_L7968352277243653188sn_c_d @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_261_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_nat,A: c > nat > assn] :
      ( ( ( size_size_list_c @ L )
       != ( size_size_list_nat @ Li ) )
     => ( ( vEBT_L4839309063161974733_c_nat @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_262_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_int,A: c > int > assn] :
      ( ( ( size_size_list_c @ L )
       != ( size_size_list_int @ Li ) )
     => ( ( vEBT_L4836818592652924457_c_int @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_263_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_c,A: d > c > assn] :
      ( ( ( size_size_list_d @ L )
       != ( size_size_list_c @ Li ) )
     => ( ( vEBT_L5180896695270977154sn_d_c @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_264_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_d,A: d > d > assn] :
      ( ( ( size_size_list_d @ L )
       != ( size_size_list_d @ Li ) )
     => ( ( vEBT_L5180896695270977155sn_d_d @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_265_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_nat,A: d > nat > assn] :
      ( ( ( size_size_list_d @ L )
       != ( size_size_list_nat @ Li ) )
     => ( ( vEBT_L6074753392118369230_d_nat @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_266_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_int,A: d > int > assn] :
      ( ( ( size_size_list_d @ L )
       != ( size_size_list_int @ Li ) )
     => ( ( vEBT_L6072262921609318954_d_int @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_267_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li: list_c,A: nat > c > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_c @ Li ) )
     => ( ( vEBT_L6400351906427472171_nat_c @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_268_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li: list_d,A: nat > d > assn] :
      ( ( ( size_size_list_nat @ L )
       != ( size_size_list_d @ Li ) )
     => ( ( vEBT_L6400351906427472172_nat_d @ A @ L @ Li )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_269_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_c,X: c,X6: c] :
      ( ( I != I2 )
     => ( ( list_update_c @ ( list_update_c @ Xs @ I @ X ) @ I2 @ X6 )
        = ( list_update_c @ ( list_update_c @ Xs @ I2 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_270_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_d,X: d,X6: d] :
      ( ( I != I2 )
     => ( ( list_update_d @ ( list_update_d @ Xs @ I @ X ) @ I2 @ X6 )
        = ( list_update_d @ ( list_update_d @ Xs @ I2 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_271_insert__Diff__single,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A2 @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_272_insert__Diff__single,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_273_insert__Diff__single,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o2 @ A2 @ ( minus_minus_set_o @ A @ ( insert_o2 @ A2 @ bot_bot_set_o ) ) )
      = ( insert_o2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_274_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int2 @ A2 @ ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ bot_bot_set_int ) ) )
      = ( insert_int2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_275_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_276_insert__Diff1,axiom,
    ! [X: $o,B: set_o,A: set_o] :
      ( ( member_o @ X @ B )
     => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A ) @ B )
        = ( minus_minus_set_o @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_277_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ B )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ B )
        = ( minus_3314409938677909166at_nat @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_278_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ B )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ B )
        = ( minus_1356011639430497352at_nat @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_279_insert__Diff1,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( ( member_int @ X @ B )
     => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B )
        = ( minus_minus_set_int @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_280_insert__Diff1,axiom,
    ! [X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ B )
     => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B )
        = ( minus_1801376950450012436_nat_o @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_281_insert__Diff1,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( ( member_nat @ X @ B )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
        = ( minus_minus_set_nat @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_282_Diff__insert0,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( minus_minus_set_o @ A @ ( insert_o2 @ X @ B ) )
        = ( minus_minus_set_o @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_283_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A )
     => ( ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
        = ( minus_3314409938677909166at_nat @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_284_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A )
     => ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
        = ( minus_1356011639430497352at_nat @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_285_Diff__insert0,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( minus_minus_set_int @ A @ ( insert_int2 @ X @ B ) )
        = ( minus_minus_set_int @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_286_Diff__insert0,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A )
     => ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
        = ( minus_1801376950450012436_nat_o @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_287_Diff__insert0,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ B ) )
        = ( minus_minus_set_nat @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_288_Diff__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% Diff_empty
thf(fact_289_Diff__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% Diff_empty
thf(fact_290_Diff__empty,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Diff_empty
thf(fact_291_Diff__empty,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Diff_empty
thf(fact_292_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_293_empty__Diff,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ bot_bo228742789529271731at_nat @ A )
      = bot_bo228742789529271731at_nat ) ).

% empty_Diff
thf(fact_294_empty__Diff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_295_empty__Diff,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_296_empty__Diff,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_297_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_298_Diff__cancel,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ A )
      = bot_bo228742789529271731at_nat ) ).

% Diff_cancel
thf(fact_299_Diff__cancel,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ A )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_300_Diff__cancel,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ A )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_301_Diff__cancel,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ A )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_302_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_303_singletonI,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).

% singletonI
thf(fact_304_singletonI,axiom,
    ! [A2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_305_singletonI,axiom,
    ! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_306_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o2 @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_307_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_308_singletonI,axiom,
    ! [A2: int] : ( member_int @ A2 @ ( insert_int2 @ A2 @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_309_Diff__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_310_Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_311_Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o2 @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o2 @ A2 @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_312_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B ) @ ( insert_int2 @ A2 @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_313_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_314_insert__Diff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_315_insert__Diff,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ A )
     => ( ( insert9069300056098147895at_nat @ A2 @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_316_insert__Diff,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A )
     => ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_317_insert__Diff,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o2 @ A2 @ ( minus_minus_set_o @ A @ ( insert_o2 @ A2 @ bot_bot_set_o ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_318_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int2 @ A2 @ ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ bot_bot_set_int ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_319_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_320_Diff__insert2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_321_Diff__insert2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_322_Diff__insert2,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o2 @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o2 @ A2 @ bot_bot_set_o ) ) @ B ) ) ).

% Diff_insert2
thf(fact_323_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int2 @ A2 @ bot_bot_set_int ) ) @ B ) ) ).

% Diff_insert2
thf(fact_324_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_325_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_326_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_327_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A ) @ ( insert_o2 @ Y @ bot_bot_set_o ) )
        = ( insert_o2 @ X @ ( minus_minus_set_o @ A @ ( insert_o2 @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_328_insert__minus__eq,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ ( insert_int2 @ Y @ bot_bot_set_int ) )
        = ( insert_int2 @ X @ ( minus_minus_set_int @ A @ ( insert_int2 @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_329_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ ( insert_nat2 @ Y @ bot_bot_set_nat ) )
        = ( insert_nat2 @ X @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_330_Diff__insert__absorb,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A )
     => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_331_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_332_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_333_Diff__insert__absorb,axiom,
    ! [X: $o,A: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A ) @ ( insert_o2 @ X @ bot_bot_set_o ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_334_Diff__insert__absorb,axiom,
    ! [X: int,A: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ ( insert_int2 @ X @ bot_bot_set_int ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_335_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_336_empty__Collect__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( collec939566748876313656_nat_o @ P ) )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_337_empty__Collect__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( bot_bo228742789529271731at_nat
        = ( collec6321179662152712658at_nat @ P ) )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_338_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X2: product_prod_nat_nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_339_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_340_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_341_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_342_Collect__empty__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_343_Collect__empty__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( ( collec6321179662152712658at_nat @ P )
        = bot_bo228742789529271731at_nat )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_344_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_345_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_346_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_347_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_348_all__not__in__conv,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X2 @ A ) )
      = ( A = bot_bo7824918357723582233_nat_o ) ) ).

% all_not_in_conv
thf(fact_349_all__not__in__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ! [X2: produc3843707927480180839at_nat] :
            ~ ( member8757157785044589968at_nat @ X2 @ A ) )
      = ( A = bot_bo228742789529271731at_nat ) ) ).

% all_not_in_conv
thf(fact_350_all__not__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ! [X2: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X2 @ A ) )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_351_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X2: $o] :
            ~ ( member_o @ X2 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_352_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat @ X2 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_353_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ( ! [X2: int] :
            ~ ( member_int @ X2 @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_354_empty__iff,axiom,
    ! [C: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ C @ bot_bo7824918357723582233_nat_o ) ).

% empty_iff
thf(fact_355_empty__iff,axiom,
    ! [C: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ C @ bot_bo228742789529271731at_nat ) ).

% empty_iff
thf(fact_356_empty__iff,axiom,
    ! [C: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_357_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_358_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_359_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_360_insertCI,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( ~ ( member_o @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_o @ A2 @ ( insert_o2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_361_insertCI,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_362_insertCI,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_363_insertCI,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_364_insertCI,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( ~ ( member_int @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_int @ A2 @ ( insert_int2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_365_insertCI,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o] :
      ( ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B2 @ B ) ) ) ).

% insertCI
thf(fact_366_insert__iff,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o2 @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_367_insert__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member8757157785044589968at_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_368_insert__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member8440522571783428010at_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_369_insert__iff,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat2 @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_370_insert__iff,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( ( member_int @ A2 @ ( insert_int2 @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_int @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_371_insert__iff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_372_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A ) )
      = ( insert_nat2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_373_insert__absorb2,axiom,
    ! [X: int,A: set_int] :
      ( ( insert_int2 @ X @ ( insert_int2 @ X @ A ) )
      = ( insert_int2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_374_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A ) )
      = ( insert8211810215607154385at_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_375_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ X @ A ) )
      = ( insert9069300056098147895at_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_376_insert__absorb2,axiom,
    ! [X: $o,A: set_o] :
      ( ( insert_o2 @ X @ ( insert_o2 @ X @ A ) )
      = ( insert_o2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_377_DiffI,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A )
     => ( ~ ( member8757157785044589968at_nat @ C @ B )
       => ( member8757157785044589968at_nat @ C @ ( minus_3314409938677909166at_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_378_DiffI,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A )
     => ( ~ ( member8440522571783428010at_nat @ C @ B )
       => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_379_DiffI,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ A )
     => ( ~ ( member_int @ C @ B )
       => ( member_int @ C @ ( minus_minus_set_int @ A @ B ) ) ) ) ).

% DiffI
thf(fact_380_DiffI,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ A )
     => ( ~ ( member6576561426505652726_nat_o @ C @ B )
       => ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B ) ) ) ) ).

% DiffI
thf(fact_381_DiffI,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A )
     => ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_382_Diff__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( minus_3314409938677909166at_nat @ A @ B ) )
      = ( ( member8757157785044589968at_nat @ C @ A )
        & ~ ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_383_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = ( ( member8440522571783428010at_nat @ C @ A )
        & ~ ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_384_Diff__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
      = ( ( member_int @ C @ A )
        & ~ ( member_int @ C @ B ) ) ) ).

% Diff_iff
thf(fact_385_Diff__iff,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B ) )
      = ( ( member6576561426505652726_nat_o @ C @ A )
        & ~ ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% Diff_iff
thf(fact_386_Diff__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
      = ( ( member_nat @ C @ A )
        & ~ ( member_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_387_Diff__idemp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
      = ( minus_minus_set_nat @ A @ B ) ) ).

% Diff_idemp
thf(fact_388_not__psubset__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% not_psubset_empty
thf(fact_389_not__psubset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_390_not__psubset__empty,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_391_not__psubset__empty,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_392_not__psubset__empty,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_393_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A @ B )
     => ? [B4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ B4 @ ( minus_3314409938677909166at_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_394_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ B )
     => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_395_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_396_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2453136405763929_nat_o @ A @ B )
     => ? [B4: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ B4 @ ( minus_1801376950450012436_nat_o @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_397_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_398_remove__subset,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).

% remove_subset
thf(fact_399_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ S )
     => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_400_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ S )
     => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_401_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ S ) ) ).

% remove_subset
thf(fact_402_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( ( member_int @ X @ S )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ S ) ) ).

% remove_subset
thf(fact_403_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat @ X @ S )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_404_memb__imp__not__empty,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( S != bot_bo7824918357723582233_nat_o ) ) ).

% memb_imp_not_empty
thf(fact_405_memb__imp__not__empty,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ S )
     => ( S != bot_bo228742789529271731at_nat ) ) ).

% memb_imp_not_empty
thf(fact_406_memb__imp__not__empty,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
thf(fact_407_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_408_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat @ X @ S )
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_409_memb__imp__not__empty,axiom,
    ! [X: int,S: set_int] :
      ( ( member_int @ X @ S )
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_410_set__notEmptyE,axiom,
    ! [S: set_Pr4532377907799695533_nat_o] :
      ( ( S != bot_bo7824918357723582233_nat_o )
     => ~ ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_411_set__notEmptyE,axiom,
    ! [S: set_Pr4329608150637261639at_nat] :
      ( ( S != bot_bo228742789529271731at_nat )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ~ ( member8757157785044589968at_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_412_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_413_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_414_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_415_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_416_bot__set__def,axiom,
    ( bot_bo7824918357723582233_nat_o
    = ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).

% bot_set_def
thf(fact_417_bot__set__def,axiom,
    ( bot_bo228742789529271731at_nat
    = ( collec6321179662152712658at_nat @ bot_bo6829958733157050602_nat_o ) ) ).

% bot_set_def
thf(fact_418_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_419_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_420_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_421_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_422_ex__in__conv,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
      = ( A != bot_bo7824918357723582233_nat_o ) ) ).

% ex_in_conv
thf(fact_423_ex__in__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ? [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ A ) )
      = ( A != bot_bo228742789529271731at_nat ) ) ).

% ex_in_conv
thf(fact_424_ex__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ? [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A ) )
      = ( A != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_425_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_426_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_427_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ( ? [X2: int] : ( member_int @ X2 @ A ) )
      = ( A != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_428_equals0I,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ! [Y2: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ Y2 @ A )
     => ( A = bot_bo7824918357723582233_nat_o ) ) ).

% equals0I
thf(fact_429_equals0I,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ! [Y2: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ Y2 @ A )
     => ( A = bot_bo228742789529271731at_nat ) ) ).

% equals0I
thf(fact_430_equals0I,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [Y2: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y2 @ A )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_431_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y2: $o] :
          ~ ( member_o @ Y2 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_432_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y2: nat] :
          ~ ( member_nat @ Y2 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_433_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y2: int] :
          ~ ( member_int @ Y2 @ A )
     => ( A = bot_bot_set_int ) ) ).

% equals0I
thf(fact_434_equals0D,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
      ( ( A = bot_bo7824918357723582233_nat_o )
     => ~ ( member6576561426505652726_nat_o @ A2 @ A ) ) ).

% equals0D
thf(fact_435_equals0D,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ( A = bot_bo228742789529271731at_nat )
     => ~ ( member8757157785044589968at_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_436_equals0D,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( A = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_437_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_438_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_439_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ ( member_int @ A2 @ A ) ) ).

% equals0D
thf(fact_440_emptyE,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).

% emptyE
thf(fact_441_emptyE,axiom,
    ! [A2: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ A2 @ bot_bo228742789529271731at_nat ) ).

% emptyE
thf(fact_442_emptyE,axiom,
    ! [A2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_443_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_444_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_445_emptyE,axiom,
    ! [A2: int] :
      ~ ( member_int @ A2 @ bot_bot_set_int ) ).

% emptyE
thf(fact_446_insertE,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o2 @ B2 @ A ) )
     => ( ( A2 = ~ B2 )
       => ( member_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_447_insertE,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member8757157785044589968at_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_448_insertE,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member8440522571783428010at_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_449_insertE,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat2 @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_450_insertE,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( ( member_int @ A2 @ ( insert_int2 @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_int @ A2 @ A ) ) ) ).

% insertE
thf(fact_451_insertE,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_452_insertI1,axiom,
    ! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o2 @ A2 @ B ) ) ).

% insertI1
thf(fact_453_insertI1,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_454_insertI1,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_455_insertI1,axiom,
    ! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).

% insertI1
thf(fact_456_insertI1,axiom,
    ! [A2: int,B: set_int] : ( member_int @ A2 @ ( insert_int2 @ A2 @ B ) ) ).

% insertI1
thf(fact_457_insertI1,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ B ) ) ).

% insertI1
thf(fact_458_insertI2,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( member_o @ A2 @ B )
     => ( member_o @ A2 @ ( insert_o2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_459_insertI2,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ B )
     => ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_460_insertI2,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ B )
     => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_461_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A2 @ B )
     => ( member_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_462_insertI2,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( member_int @ A2 @ B )
     => ( member_int @ A2 @ ( insert_int2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_463_insertI2,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ A2 @ B )
     => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B2 @ B ) ) ) ).

% insertI2
thf(fact_464_Set_Oset__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( member_o @ X @ A )
     => ~ ! [B5: set_o] :
            ( ( A
              = ( insert_o2 @ X @ B5 ) )
           => ( member_o @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_465_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A )
     => ~ ! [B5: set_Pr4329608150637261639at_nat] :
            ( ( A
              = ( insert9069300056098147895at_nat @ X @ B5 ) )
           => ( member8757157785044589968at_nat @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_466_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A )
     => ~ ! [B5: set_Pr1261947904930325089at_nat] :
            ( ( A
              = ( insert8211810215607154385at_nat @ X @ B5 ) )
           => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_467_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ~ ! [B5: set_nat] :
            ( ( A
              = ( insert_nat2 @ X @ B5 ) )
           => ( member_nat @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_468_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( member_int @ X @ A )
     => ~ ! [B5: set_int] :
            ( ( A
              = ( insert_int2 @ X @ B5 ) )
           => ( member_int @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_469_Set_Oset__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ A )
     => ~ ! [B5: set_Pr4532377907799695533_nat_o] :
            ( ( A
              = ( insert5175938949040314269_nat_o @ X @ B5 ) )
           => ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_470_insert__ident,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ~ ( member_o @ X @ B )
       => ( ( ( insert_o2 @ X @ A )
            = ( insert_o2 @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_471_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A )
     => ( ~ ( member8757157785044589968at_nat @ X @ B )
       => ( ( ( insert9069300056098147895at_nat @ X @ A )
            = ( insert9069300056098147895at_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_472_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A )
     => ( ~ ( member8440522571783428010at_nat @ X @ B )
       => ( ( ( insert8211810215607154385at_nat @ X @ A )
            = ( insert8211810215607154385at_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_473_insert__ident,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ~ ( member_nat @ X @ B )
       => ( ( ( insert_nat2 @ X @ A )
            = ( insert_nat2 @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_474_insert__ident,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ~ ( member_int @ X @ B )
       => ( ( ( insert_int2 @ X @ A )
            = ( insert_int2 @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_475_insert__ident,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A )
     => ( ~ ( member6576561426505652726_nat_o @ X @ B )
       => ( ( ( insert5175938949040314269_nat_o @ X @ A )
            = ( insert5175938949040314269_nat_o @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_476_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_477_insert__absorb,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ A )
     => ( ( insert9069300056098147895at_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_478_insert__absorb,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A )
     => ( ( insert8211810215607154385at_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_479_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_480_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_481_insert__absorb,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ( insert5175938949040314269_nat_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_482_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B2: $o,B: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ~ ( member_o @ B2 @ B )
       => ( ( ( insert_o2 @ A2 @ A )
            = ( insert_o2 @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 = ~ B2 )
             => ? [C3: set_o] :
                  ( ( A
                    = ( insert_o2 @ B2 @ C3 ) )
                  & ~ ( member_o @ B2 @ C3 )
                  & ( B
                    = ( insert_o2 @ A2 @ C3 ) )
                  & ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_483_insert__eq__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A2 @ A )
     => ( ~ ( member8757157785044589968at_nat @ B2 @ B )
       => ( ( ( insert9069300056098147895at_nat @ A2 @ A )
            = ( insert9069300056098147895at_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_Pr4329608150637261639at_nat] :
                  ( ( A
                    = ( insert9069300056098147895at_nat @ B2 @ C3 ) )
                  & ~ ( member8757157785044589968at_nat @ B2 @ C3 )
                  & ( B
                    = ( insert9069300056098147895at_nat @ A2 @ C3 ) )
                  & ~ ( member8757157785044589968at_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_484_insert__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ A )
     => ( ~ ( member8440522571783428010at_nat @ B2 @ B )
       => ( ( ( insert8211810215607154385at_nat @ A2 @ A )
            = ( insert8211810215607154385at_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_Pr1261947904930325089at_nat] :
                  ( ( A
                    = ( insert8211810215607154385at_nat @ B2 @ C3 ) )
                  & ~ ( member8440522571783428010at_nat @ B2 @ C3 )
                  & ( B
                    = ( insert8211810215607154385at_nat @ A2 @ C3 ) )
                  & ~ ( member8440522571783428010at_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_485_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B2 @ B )
       => ( ( ( insert_nat2 @ A2 @ A )
            = ( insert_nat2 @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_nat] :
                  ( ( A
                    = ( insert_nat2 @ B2 @ C3 ) )
                  & ~ ( member_nat @ B2 @ C3 )
                  & ( B
                    = ( insert_nat2 @ A2 @ C3 ) )
                  & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_486_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B2: int,B: set_int] :
      ( ~ ( member_int @ A2 @ A )
     => ( ~ ( member_int @ B2 @ B )
       => ( ( ( insert_int2 @ A2 @ A )
            = ( insert_int2 @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_int] :
                  ( ( A
                    = ( insert_int2 @ B2 @ C3 ) )
                  & ~ ( member_int @ B2 @ C3 )
                  & ( B
                    = ( insert_int2 @ A2 @ C3 ) )
                  & ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_487_insert__eq__iff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ~ ( member6576561426505652726_nat_o @ B2 @ B )
       => ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
            = ( insert5175938949040314269_nat_o @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_Pr4532377907799695533_nat_o] :
                  ( ( A
                    = ( insert5175938949040314269_nat_o @ B2 @ C3 ) )
                  & ~ ( member6576561426505652726_nat_o @ B2 @ C3 )
                  & ( B
                    = ( insert5175938949040314269_nat_o @ A2 @ C3 ) )
                  & ~ ( member6576561426505652726_nat_o @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_488_insert__commute,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( insert_nat2 @ X @ ( insert_nat2 @ Y @ A ) )
      = ( insert_nat2 @ Y @ ( insert_nat2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_489_insert__commute,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( insert_int2 @ X @ ( insert_int2 @ Y @ A ) )
      = ( insert_int2 @ Y @ ( insert_int2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_490_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A ) )
      = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A ) ) ) ).

% insert_commute
thf(fact_491_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ A ) )
      = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X @ A ) ) ) ).

% insert_commute
thf(fact_492_insert__commute,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( insert_o2 @ X @ ( insert_o2 @ Y @ A ) )
      = ( insert_o2 @ Y @ ( insert_o2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_493_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B5: set_o] :
          ( ( A
            = ( insert_o2 @ A2 @ B5 ) )
          & ~ ( member_o @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_494_mk__disjoint__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ A )
     => ? [B5: set_Pr4329608150637261639at_nat] :
          ( ( A
            = ( insert9069300056098147895at_nat @ A2 @ B5 ) )
          & ~ ( member8757157785044589968at_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_495_mk__disjoint__insert,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A )
     => ? [B5: set_Pr1261947904930325089at_nat] :
          ( ( A
            = ( insert8211810215607154385at_nat @ A2 @ B5 ) )
          & ~ ( member8440522571783428010at_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_496_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B5: set_nat] :
          ( ( A
            = ( insert_nat2 @ A2 @ B5 ) )
          & ~ ( member_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_497_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ? [B5: set_int] :
          ( ( A
            = ( insert_int2 @ A2 @ B5 ) )
          & ~ ( member_int @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_498_mk__disjoint__insert,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ A )
     => ? [B5: set_Pr4532377907799695533_nat_o] :
          ( ( A
            = ( insert5175938949040314269_nat_o @ A2 @ B5 ) )
          & ~ ( member6576561426505652726_nat_o @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_499_DiffE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( minus_3314409938677909166at_nat @ A @ B ) )
     => ~ ( ( member8757157785044589968at_nat @ C @ A )
         => ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_500_DiffE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A )
         => ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_501_DiffE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( ( member_int @ C @ A )
         => ( member_int @ C @ B ) ) ) ).

% DiffE
thf(fact_502_DiffE,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B ) )
     => ~ ( ( member6576561426505652726_nat_o @ C @ A )
         => ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% DiffE
thf(fact_503_DiffE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( ( member_nat @ C @ A )
         => ( member_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_504_DiffD1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( minus_3314409938677909166at_nat @ A @ B ) )
     => ( member8757157785044589968at_nat @ C @ A ) ) ).

% DiffD1
thf(fact_505_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B ) )
     => ( member8440522571783428010at_nat @ C @ A ) ) ).

% DiffD1
thf(fact_506_DiffD1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ( member_int @ C @ A ) ) ).

% DiffD1
thf(fact_507_DiffD1,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B ) )
     => ( member6576561426505652726_nat_o @ C @ A ) ) ).

% DiffD1
thf(fact_508_DiffD1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ( member_nat @ C @ A ) ) ).

% DiffD1
thf(fact_509_DiffD2,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( minus_3314409938677909166at_nat @ A @ B ) )
     => ~ ( member8757157785044589968at_nat @ C @ B ) ) ).

% DiffD2
thf(fact_510_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A @ B ) )
     => ~ ( member8440522571783428010at_nat @ C @ B ) ) ).

% DiffD2
thf(fact_511_DiffD2,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( member_int @ C @ B ) ) ).

% DiffD2
thf(fact_512_DiffD2,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B ) )
     => ~ ( member6576561426505652726_nat_o @ C @ B ) ) ).

% DiffD2
thf(fact_513_DiffD2,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( member_nat @ C @ B ) ) ).

% DiffD2
thf(fact_514_singleton__inject,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_515_singleton__inject,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_516_singleton__inject,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( insert_o2 @ A2 @ bot_bot_set_o )
        = ( insert_o2 @ B2 @ bot_bot_set_o ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_517_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
        = ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_518_singleton__inject,axiom,
    ! [A2: int,B2: int] :
      ( ( ( insert_int2 @ A2 @ bot_bot_set_int )
        = ( insert_int2 @ B2 @ bot_bot_set_int ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_519_insert__not__empty,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A2 @ A )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_520_insert__not__empty,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ A )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_521_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o2 @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_522_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat2 @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_523_insert__not__empty,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int2 @ A2 @ A )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_524_doubleton__eq__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D2 @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_525_doubleton__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_526_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C: $o,D2: $o] :
      ( ( ( insert_o2 @ A2 @ ( insert_o2 @ B2 @ bot_bot_set_o ) )
        = ( insert_o2 @ C @ ( insert_o2 @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_527_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_528_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( insert_int2 @ A2 @ ( insert_int2 @ B2 @ bot_bot_set_int ) )
        = ( insert_int2 @ C @ ( insert_int2 @ D2 @ bot_bot_set_int ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_529_singleton__iff,axiom,
    ! [B2: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_530_singleton__iff,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_531_singleton__iff,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_532_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o2 @ A2 @ bot_bot_set_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_533_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_534_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int2 @ A2 @ bot_bot_set_int ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_535_singletonD,axiom,
    ! [B2: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_536_singletonD,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_537_singletonD,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_538_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o2 @ A2 @ bot_bot_set_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_539_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_540_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int2 @ A2 @ bot_bot_set_int ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_541_insert__Diff__if,axiom,
    ! [X: $o,B: set_o,A: set_o] :
      ( ( ( member_o @ X @ B )
       => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A ) @ B )
          = ( minus_minus_set_o @ A @ B ) ) )
      & ( ~ ( member_o @ X @ B )
       => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A ) @ B )
          = ( insert_o2 @ X @ ( minus_minus_set_o @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_542_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ B )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ B )
          = ( minus_3314409938677909166at_nat @ A @ B ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ B )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ B )
          = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_543_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ B )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ B )
          = ( minus_1356011639430497352at_nat @ A @ B ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ B )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ B )
          = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_544_insert__Diff__if,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( ( ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B )
          = ( minus_minus_set_int @ A @ B ) ) )
      & ( ~ ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A ) @ B )
          = ( insert_int2 @ X @ ( minus_minus_set_int @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_545_insert__Diff__if,axiom,
    ! [X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ X @ B )
       => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B )
          = ( minus_1801376950450012436_nat_o @ A @ B ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ X @ B )
       => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B )
          = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_546_insert__Diff__if,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( ( ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
          = ( minus_minus_set_nat @ A @ B ) ) )
      & ( ~ ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B )
          = ( insert_nat2 @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_547_set__minus__singleton__eq,axiom,
    ! [X: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ X4 )
     => ( ( minus_1801376950450012436_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_548_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ X4 )
     => ( ( minus_3314409938677909166at_nat @ X4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_549_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ X4 )
     => ( ( minus_1356011639430497352at_nat @ X4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_550_set__minus__singleton__eq,axiom,
    ! [X: $o,X4: set_o] :
      ( ~ ( member_o @ X @ X4 )
     => ( ( minus_minus_set_o @ X4 @ ( insert_o2 @ X @ bot_bot_set_o ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_551_set__minus__singleton__eq,axiom,
    ! [X: int,X4: set_int] :
      ( ~ ( member_int @ X @ X4 )
     => ( ( minus_minus_set_int @ X4 @ ( insert_int2 @ X @ bot_bot_set_int ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_552_set__minus__singleton__eq,axiom,
    ! [X: nat,X4: set_nat] :
      ( ~ ( member_nat @ X @ X4 )
     => ( ( minus_minus_set_nat @ X4 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = X4 ) ) ).

% set_minus_singleton_eq
thf(fact_553_listI__assn__reinsert,axiom,
    ! [P: assn,A: c > nat > assn,Xs: list_c,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L6585172037124793648_c_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6585172037124793648_c_nat @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_554_listI__assn__reinsert,axiom,
    ! [P: assn,A: c > int > assn,Xs: list_c,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L6582681566615743372_c_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6582681566615743372_c_int @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_555_listI__assn__reinsert,axiom,
    ! [P: assn,A: d > nat > assn,Xs: list_d,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7820616366081188145_d_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7820616366081188145_d_nat @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_556_listI__assn__reinsert,axiom,
    ! [P: assn,A: d > int > assn,Xs: list_d,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7818125895572137869_d_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7818125895572137869_d_int @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_557_listI__assn__reinsert,axiom,
    ! [P: assn,A: nat > nat > assn,Xs: list_nat,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7892069108621897922at_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7892069108621897922at_nat @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_558_listI__assn__reinsert,axiom,
    ! [P: assn,A: nat > int > assn,Xs: list_nat,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7889578638112847646at_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7889578638112847646at_int @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_559_listI__assn__reinsert,axiom,
    ! [P: assn,A: int > nat > assn,Xs: list_int,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8891422820522952478nt_nat @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_560_listI__assn__reinsert,axiom,
    ! [P: assn,A: int > int > assn,Xs: list_int,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8888932350013902202nt_int @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_561_listI__assn__reinsert,axiom,
    ! [P: assn,A: c > d > assn,Xs: list_c,I: nat,Xsi: list_d,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_d @ Xsi @ I ) ) @ ( vEBT_L824818263033719201sn_c_d @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L824818263033719201sn_c_d @ I3 @ A @ Xs @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_562_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_563_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_564_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o2 @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_565_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_566_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int2 @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_567_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L6585172037124793648_c_nat @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L6585172037124793648_c_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_568_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L6582681566615743372_c_int @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L6582681566615743372_c_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_569_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: d > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( vEBT_L7820616366081188145_d_nat @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7820616366081188145_d_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_570_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: d > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( vEBT_L7818125895572137869_d_int @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7818125895572137869_d_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_571_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: nat > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7892069108621897922at_nat @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7892069108621897922at_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_572_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: nat > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7889578638112847646at_int @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7889578638112847646at_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_573_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: int > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8891422820522952478nt_nat @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_574_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: int > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8888932350013902202nt_int @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_575_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > d > assn,Xsi: list_d] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L824818263033719201sn_c_d @ I3 @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_d @ Xsi @ I ) ) @ ( vEBT_L824818263033719201sn_c_d @ ( minus_minus_set_nat @ I3 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_576_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_577_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_578_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o2 @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_579_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_580_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int2 @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_581_inf__period_I2_J,axiom,
    ! [P: code_integer > $o,D: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K @ D ) ) ) )
     => ( ! [X3: code_integer,K: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K @ D ) ) ) )
       => ! [X7: code_integer,K2: code_integer] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K2 @ D ) ) )
              | ( Q @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_582_inf__period_I2_J,axiom,
    ! [P: rat > $o,D: rat,Q: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D ) ) ) )
       => ! [X7: rat,K2: rat] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K2 @ D ) ) )
              | ( Q @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_583_inf__period_I2_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
       => ! [X7: int,K2: int] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K2 @ D ) ) )
              | ( Q @ ( minus_minus_int @ X7 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_584_inf__period_I1_J,axiom,
    ! [P: code_integer > $o,D: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K @ D ) ) ) )
     => ( ! [X3: code_integer,K: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K @ D ) ) ) )
       => ! [X7: code_integer,K2: code_integer] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K2 @ D ) ) )
              & ( Q @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_585_inf__period_I1_J,axiom,
    ! [P: rat > $o,D: rat,Q: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D ) ) ) )
       => ! [X7: rat,K2: rat] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K2 @ D ) ) )
              & ( Q @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_586_inf__period_I1_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
       => ! [X7: int,K2: int] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K2 @ D ) ) )
              & ( Q @ ( minus_minus_int @ X7 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_587_left__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_588_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_589_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_590_right__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_591_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_592_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_593_left__diff__distrib_H,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ C ) @ A2 )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ ( times_3573771949741848930nteger @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_594_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A2 )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A2 ) @ ( times_times_rat @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_595_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A2 )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_596_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A2 )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_597_right__diff__distrib_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_598_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_599_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_600_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_601_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ D2 @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_602_diff__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,D2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ D2 @ C )
       => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_603_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_604_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_605_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
        = ( nth_int @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_606_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_c,X: c] :
      ( ( I != J )
     => ( ( nth_c @ ( list_update_c @ Xs @ I @ X ) @ J )
        = ( nth_c @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_607_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_d,X: d] :
      ( ( I != J )
     => ( ( nth_d @ ( list_update_d @ Xs @ I @ X ) @ J )
        = ( nth_d @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_608_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_609_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_610_list__update__id,axiom,
    ! [Xs: list_c,I: nat] :
      ( ( list_update_c @ Xs @ I @ ( nth_c @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_611_list__update__id,axiom,
    ! [Xs: list_d,I: nat] :
      ( ( list_update_d @ Xs @ I @ ( nth_d @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_612_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_c,X: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( nth_c @ ( list_update_c @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_613_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_d,X: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( nth_d @ ( list_update_d @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_614_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_615_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_616_nth__update__invalid,axiom,
    ! [I: nat,L: list_c,J: nat,X: c] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_c @ L ) )
     => ( ( nth_c @ ( list_update_c @ L @ J @ X ) @ I )
        = ( nth_c @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_617_nth__update__invalid,axiom,
    ! [I: nat,L: list_d,J: nat,X: d] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_d @ L ) )
     => ( ( nth_d @ ( list_update_d @ L @ J @ X ) @ I )
        = ( nth_d @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_618_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( list_update_nat @ L @ J @ X ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_619_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( list_update_int @ L @ J @ X ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_620_psubsetD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A @ B )
     => ( ( member8757157785044589968at_nat @ C @ A )
       => ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_621_psubsetD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ B )
     => ( ( member8440522571783428010at_nat @ C @ A )
       => ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_622_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_623_psubsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% psubsetD
thf(fact_624_psubsetD,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2453136405763929_nat_o @ A @ B )
     => ( ( member6576561426505652726_nat_o @ C @ A )
       => ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% psubsetD
thf(fact_625_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_c,Z2: list_c] : Y4 = Z2 )
    = ( ^ [Xs3: list_c,Ys3: list_c] :
          ( ( ( size_size_list_c @ Xs3 )
            = ( size_size_list_c @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_c @ Xs3 ) )
             => ( ( nth_c @ Xs3 @ I4 )
                = ( nth_c @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_626_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_d,Z2: list_d] : Y4 = Z2 )
    = ( ^ [Xs3: list_d,Ys3: list_d] :
          ( ( ( size_size_list_d @ Xs3 )
            = ( size_size_list_d @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_d @ Xs3 ) )
             => ( ( nth_d @ Xs3 @ I4 )
                = ( nth_d @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_627_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2 )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_628_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_int,Z2: list_int] : Y4 = Z2 )
    = ( ^ [Xs3: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_629_Skolem__list__nth,axiom,
    ! [K3: nat,P: nat > c > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K3 )
           => ? [X8: c] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs3: list_c] :
            ( ( ( size_size_list_c @ Xs3 )
              = K3 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ( P @ I4 @ ( nth_c @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_630_Skolem__list__nth,axiom,
    ! [K3: nat,P: nat > d > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K3 )
           => ? [X8: d] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs3: list_d] :
            ( ( ( size_size_list_d @ Xs3 )
              = K3 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ( P @ I4 @ ( nth_d @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_631_Skolem__list__nth,axiom,
    ! [K3: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K3 )
           => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K3 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_632_Skolem__list__nth,axiom,
    ! [K3: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K3 )
           => ? [X8: int] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K3 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ( P @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_633_nth__equalityI,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
           => ( ( nth_c @ Xs @ I5 )
              = ( nth_c @ Ys @ I5 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_634_nth__equalityI,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
           => ( ( nth_d @ Xs @ I5 )
              = ( nth_d @ Ys @ I5 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_635_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I5 )
              = ( nth_nat @ Ys @ I5 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_636_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I5 )
              = ( nth_int @ Ys @ I5 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_637_obtain__list__from__elements,axiom,
    ! [N: nat,P: c > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ? [Li2: c] : ( P @ Li2 @ I5 ) )
     => ~ ! [L2: list_c] :
            ( ( ( size_size_list_c @ L2 )
              = N )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N )
                 => ( P @ ( nth_c @ L2 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_638_obtain__list__from__elements,axiom,
    ! [N: nat,P: d > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ? [Li2: d] : ( P @ Li2 @ I5 ) )
     => ~ ! [L2: list_d] :
            ( ( ( size_size_list_d @ L2 )
              = N )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N )
                 => ( P @ ( nth_d @ L2 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_639_obtain__list__from__elements,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ? [Li2: nat] : ( P @ Li2 @ I5 ) )
     => ~ ! [L2: list_nat] :
            ( ( ( size_size_list_nat @ L2 )
              = N )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N )
                 => ( P @ ( nth_nat @ L2 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_640_obtain__list__from__elements,axiom,
    ! [N: nat,P: int > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ? [Li2: int] : ( P @ Li2 @ I5 ) )
     => ~ ! [L2: list_int] :
            ( ( ( size_size_list_int @ L2 )
              = N )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N )
                 => ( P @ ( nth_int @ L2 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_641_is__singleton__the__elem,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat] :
          ( A5
          = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A5 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_642_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( A5
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A5 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_643_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o2 @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_644_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat2 @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_645_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
          ( A5
          = ( insert_int2 @ ( the_elem_int @ A5 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_646_is__singletonI_H,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( A != bot_bo7824918357723582233_nat_o )
     => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ A )
           => ( ( member6576561426505652726_nat_o @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_sin5180296473474724033_nat_o @ A ) ) ) ).

% is_singletonI'
thf(fact_647_is__singletonI_H,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( A != bot_bo228742789529271731at_nat )
     => ( ! [X3: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A )
           => ( ( member8757157785044589968at_nat @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_sin2937591304547752795at_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_648_is__singletonI_H,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
     => ( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A )
           => ( ( member8440522571783428010at_nat @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_sin2850979758926227957at_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_649_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y2: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( member_o @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_650_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y2: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( member_nat @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_651_is__singletonI_H,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int,Y2: int] :
            ( ( member_int @ X3 @ A )
           => ( ( member_int @ Y2 @ A )
             => ( X3 = Y2 ) ) )
       => ( is_singleton_int @ A ) ) ) ).

% is_singletonI'
thf(fact_652_nth__list__update,axiom,
    ! [I: nat,Xs: list_c,J: nat,X: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_c @ ( list_update_c @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_c @ ( list_update_c @ Xs @ I @ X ) @ J )
            = ( nth_c @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_653_nth__list__update,axiom,
    ! [I: nat,Xs: list_d,J: nat,X: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_d @ ( list_update_d @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_d @ ( list_update_d @ Xs @ I @ X ) @ J )
            = ( nth_d @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_654_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_655_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_656_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_c,X: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ( list_update_c @ Xs @ I @ X )
          = Xs )
        = ( ( nth_c @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_657_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_d,X: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ( list_update_d @ Xs @ I @ X )
          = Xs )
        = ( ( nth_d @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_658_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_659_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_660_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_c,X: c] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_c @ L ) ) )
       => ( ( nth_c @ ( list_update_c @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_c @ L ) ) )
       => ( ( nth_c @ ( list_update_c @ L @ I @ X ) @ J )
          = ( nth_c @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_661_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_d,X: d] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_d @ L ) ) )
       => ( ( nth_d @ ( list_update_d @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_d @ L ) ) )
       => ( ( nth_d @ ( list_update_d @ L @ I @ X ) @ J )
          = ( nth_d @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_662_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J )
          = ( nth_nat @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_663_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J )
          = ( nth_int @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_664_linorder__neqE__linordered__idom,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_665_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_666_pinf_I1_J,axiom,
    ! [P: code_integer > $o,P3: code_integer > $o,Q: code_integer > $o,Q3: code_integer > $o] :
      ( ? [Z3: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: code_integer] :
          ! [X7: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_667_pinf_I1_J,axiom,
    ! [P: num > $o,P3: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X7: num] :
            ( ( ord_less_num @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_668_pinf_I1_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_669_pinf_I1_J,axiom,
    ! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X7: int] :
            ( ( ord_less_int @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_670_pinf_I2_J,axiom,
    ! [P: code_integer > $o,P3: code_integer > $o,Q: code_integer > $o,Q3: code_integer > $o] :
      ( ? [Z3: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: code_integer] :
          ! [X7: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_671_pinf_I2_J,axiom,
    ! [P: num > $o,P3: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X7: num] :
            ( ( ord_less_num @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_672_pinf_I2_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_673_pinf_I2_J,axiom,
    ! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X7: int] :
            ( ( ord_less_int @ Z4 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_674_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(3)
thf(fact_675_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(3)
thf(fact_676_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(3)
thf(fact_677_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(3)
thf(fact_678_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(4)
thf(fact_679_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(4)
thf(fact_680_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(4)
thf(fact_681_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ( X7 != T ) ) ).

% pinf(4)
thf(fact_682_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ~ ( ord_le6747313008572928689nteger @ X7 @ T ) ) ).

% pinf(5)
thf(fact_683_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ~ ( ord_less_num @ X7 @ T ) ) ).

% pinf(5)
thf(fact_684_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ~ ( ord_less_nat @ X7 @ T ) ) ).

% pinf(5)
thf(fact_685_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ~ ( ord_less_int @ X7 @ T ) ) ).

% pinf(5)
thf(fact_686_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ( ord_le6747313008572928689nteger @ T @ X7 ) ) ).

% pinf(7)
thf(fact_687_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ( ord_less_num @ T @ X7 ) ) ).

% pinf(7)
thf(fact_688_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ( ord_less_nat @ T @ X7 ) ) ).

% pinf(7)
thf(fact_689_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ( ord_less_int @ T @ X7 ) ) ).

% pinf(7)
thf(fact_690_minf_I1_J,axiom,
    ! [P: code_integer > $o,P3: code_integer > $o,Q: code_integer > $o,Q3: code_integer > $o] :
      ( ? [Z3: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: code_integer] :
          ! [X7: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_691_minf_I1_J,axiom,
    ! [P: num > $o,P3: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X7: num] :
            ( ( ord_less_num @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_692_minf_I1_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_693_minf_I1_J,axiom,
    ! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X7: int] :
            ( ( ord_less_int @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                & ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_694_minf_I2_J,axiom,
    ! [P: code_integer > $o,P3: code_integer > $o,Q: code_integer > $o,Q3: code_integer > $o] :
      ( ? [Z3: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: code_integer] :
          ! [X7: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_695_minf_I2_J,axiom,
    ! [P: num > $o,P3: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X7: num] :
            ( ( ord_less_num @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_696_minf_I2_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_697_minf_I2_J,axiom,
    ! [P: int > $o,P3: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P3 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q3 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X7: int] :
            ( ( ord_less_int @ X7 @ Z4 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P3 @ X7 )
                | ( Q3 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_698_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(3)
thf(fact_699_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(3)
thf(fact_700_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(3)
thf(fact_701_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(3)
thf(fact_702_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(4)
thf(fact_703_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(4)
thf(fact_704_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(4)
thf(fact_705_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ( X7 != T ) ) ).

% minf(4)
thf(fact_706_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ( ord_le6747313008572928689nteger @ X7 @ T ) ) ).

% minf(5)
thf(fact_707_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ( ord_less_num @ X7 @ T ) ) ).

% minf(5)
thf(fact_708_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ( ord_less_nat @ X7 @ T ) ) ).

% minf(5)
thf(fact_709_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ( ord_less_int @ X7 @ T ) ) ).

% minf(5)
thf(fact_710_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ~ ( ord_le6747313008572928689nteger @ T @ X7 ) ) ).

% minf(7)
thf(fact_711_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ~ ( ord_less_num @ T @ X7 ) ) ).

% minf(7)
thf(fact_712_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ~ ( ord_less_nat @ T @ X7 ) ) ).

% minf(7)
thf(fact_713_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ~ ( ord_less_int @ T @ X7 ) ) ).

% minf(7)
thf(fact_714_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( times_3573771949741848930nteger @ A2 @ C ) )
      = ( times_3573771949741848930nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_715_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( times_times_assn @ B2 @ ( times_times_assn @ A2 @ C ) )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_716_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C ) )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_717_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_718_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_719_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_3573771949741848930nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( times_3573771949741848930nteger @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_720_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A3: assn,B3: assn] : ( times_times_assn @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_721_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_722_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_723_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_724_mult_Oassoc,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C )
      = ( times_3573771949741848930nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_725_mult_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_726_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_727_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_728_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_729_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_c,Xsi2: list_c,A: c > c > assn,A6: c > c > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_c @ Xs4 ) )
       => ( ( ( size_size_list_c @ Xsi )
            = ( size_size_list_c @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                 => ( ( ( size_size_list_c @ Xs )
                      = ( size_size_list_c @ Xsi ) )
                   => ( ( ( nth_c @ Xs @ I5 )
                        = ( nth_c @ Xs4 @ I5 ) )
                      & ( ( nth_c @ Xsi @ I5 )
                        = ( nth_c @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_c @ Xs @ I5 ) @ ( nth_c @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_c @ Xs4 @ I5 ) @ ( nth_c @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L824818263033719200sn_c_c @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L824818263033719200sn_c_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_730_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_nat,Xsi2: list_nat,A: c > nat > assn,A6: c > nat > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_c @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                 => ( ( ( size_size_list_c @ Xs )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_c @ Xs @ I5 )
                        = ( nth_c @ Xs4 @ I5 ) )
                      & ( ( nth_nat @ Xsi @ I5 )
                        = ( nth_nat @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_c @ Xs @ I5 ) @ ( nth_nat @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_c @ Xs4 @ I5 ) @ ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L6585172037124793648_c_nat @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L6585172037124793648_c_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_731_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_int,Xsi2: list_int,A: c > int > assn,A6: c > int > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_c @ Xs4 ) )
       => ( ( ( size_size_list_int @ Xsi )
            = ( size_size_list_int @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                 => ( ( ( size_size_list_c @ Xs )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_c @ Xs @ I5 )
                        = ( nth_c @ Xs4 @ I5 ) )
                      & ( ( nth_int @ Xsi @ I5 )
                        = ( nth_int @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_c @ Xs @ I5 ) @ ( nth_int @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_c @ Xs4 @ I5 ) @ ( nth_int @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L6582681566615743372_c_int @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L6582681566615743372_c_int @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_732_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_c,Xsi2: list_c,A: d > c > assn,A6: d > c > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_d @ Xs4 ) )
       => ( ( ( size_size_list_c @ Xsi )
            = ( size_size_list_c @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                 => ( ( ( size_size_list_d @ Xs )
                      = ( size_size_list_c @ Xsi ) )
                   => ( ( ( nth_d @ Xs @ I5 )
                        = ( nth_d @ Xs4 @ I5 ) )
                      & ( ( nth_c @ Xsi @ I5 )
                        = ( nth_c @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_d @ Xs @ I5 ) @ ( nth_c @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_d @ Xs4 @ I5 ) @ ( nth_c @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7260734717915818975sn_d_c @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L7260734717915818975sn_d_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_733_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_d,Xsi2: list_d,A: d > d > assn,A6: d > d > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_d @ Xs4 ) )
       => ( ( ( size_size_list_d @ Xsi )
            = ( size_size_list_d @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                 => ( ( ( size_size_list_d @ Xs )
                      = ( size_size_list_d @ Xsi ) )
                   => ( ( ( nth_d @ Xs @ I5 )
                        = ( nth_d @ Xs4 @ I5 ) )
                      & ( ( nth_d @ Xsi @ I5 )
                        = ( nth_d @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_d @ Xs @ I5 ) @ ( nth_d @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_d @ Xs4 @ I5 ) @ ( nth_d @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7260734717915818976sn_d_d @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L7260734717915818976sn_d_d @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_734_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_nat,Xsi2: list_nat,A: d > nat > assn,A6: d > nat > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_d @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                 => ( ( ( size_size_list_d @ Xs )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_d @ Xs @ I5 )
                        = ( nth_d @ Xs4 @ I5 ) )
                      & ( ( nth_nat @ Xsi @ I5 )
                        = ( nth_nat @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_d @ Xs @ I5 ) @ ( nth_nat @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_d @ Xs4 @ I5 ) @ ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7820616366081188145_d_nat @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L7820616366081188145_d_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_735_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_int,Xsi2: list_int,A: d > int > assn,A6: d > int > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_d @ Xs4 ) )
       => ( ( ( size_size_list_int @ Xsi )
            = ( size_size_list_int @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                 => ( ( ( size_size_list_d @ Xs )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_d @ Xs @ I5 )
                        = ( nth_d @ Xs4 @ I5 ) )
                      & ( ( nth_int @ Xsi @ I5 )
                        = ( nth_int @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_d @ Xs @ I5 ) @ ( nth_int @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_d @ Xs4 @ I5 ) @ ( nth_int @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7818125895572137869_d_int @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L7818125895572137869_d_int @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_736_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_c,Xsi2: list_c,A: nat > c > assn,A6: nat > c > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Xs4 ) )
       => ( ( ( size_size_list_c @ Xsi )
            = ( size_size_list_c @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                 => ( ( ( size_size_list_nat @ Xs )
                      = ( size_size_list_c @ Xsi ) )
                   => ( ( ( nth_nat @ Xs @ I5 )
                        = ( nth_nat @ Xs4 @ I5 ) )
                      & ( ( nth_c @ Xsi @ I5 )
                        = ( nth_c @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_nat @ Xs @ I5 ) @ ( nth_c @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_nat @ Xs4 @ I5 ) @ ( nth_c @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L8146214880390291086_nat_c @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L8146214880390291086_nat_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_737_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_d,Xsi2: list_d,A: nat > d > assn,A6: nat > d > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Xs4 ) )
       => ( ( ( size_size_list_d @ Xsi )
            = ( size_size_list_d @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                 => ( ( ( size_size_list_nat @ Xs )
                      = ( size_size_list_d @ Xsi ) )
                   => ( ( ( nth_nat @ Xs @ I5 )
                        = ( nth_nat @ Xs4 @ I5 ) )
                      & ( ( nth_d @ Xsi @ I5 )
                        = ( nth_d @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_nat @ Xs @ I5 ) @ ( nth_d @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_nat @ Xs4 @ I5 ) @ ( nth_d @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L8146214880390291087_nat_d @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L8146214880390291087_nat_d @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_738_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A: nat > nat > assn,A6: nat > nat > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                 => ( ( ( size_size_list_nat @ Xs )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_nat @ Xs @ I5 )
                        = ( nth_nat @ Xs4 @ I5 ) )
                      & ( ( nth_nat @ Xsi @ I5 )
                        = ( nth_nat @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xsi @ I5 ) )
                        = ( A6 @ ( nth_nat @ Xs4 @ I5 ) @ ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7892069108621897922at_nat @ I3 @ A @ Xs @ Xsi )
              = ( vEBT_L7892069108621897922at_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_739_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: c > c > assn,A6: c > c > assn,Xs: list_c,Xs4: list_c,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_c @ Xs )
            = ( size_size_list_c @ Xs4 ) )
         => ( ( ( size_size_list_c @ Xsi )
              = ( size_size_list_c @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                   => ( ( ( size_size_list_c @ Xs )
                        = ( size_size_list_c @ Xsi ) )
                     => ( ( ( nth_c @ Xs @ I5 )
                          = ( nth_c @ Xs4 @ I5 ) )
                        & ( ( nth_c @ Xsi @ I5 )
                          = ( nth_c @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L824818263033719200sn_c_c @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L824818263033719200sn_c_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_740_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: c > nat > assn,A6: c > nat > assn,Xs: list_c,Xs4: list_c,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_c @ Xs )
            = ( size_size_list_c @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                   => ( ( ( size_size_list_c @ Xs )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_c @ Xs @ I5 )
                          = ( nth_c @ Xs4 @ I5 ) )
                        & ( ( nth_nat @ Xsi @ I5 )
                          = ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L6585172037124793648_c_nat @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L6585172037124793648_c_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_741_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: c > int > assn,A6: c > int > assn,Xs: list_c,Xs4: list_c,Xsi: list_int,Xsi2: list_int] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_c @ Xs )
            = ( size_size_list_c @ Xs4 ) )
         => ( ( ( size_size_list_int @ Xsi )
              = ( size_size_list_int @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
                   => ( ( ( size_size_list_c @ Xs )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_c @ Xs @ I5 )
                          = ( nth_c @ Xs4 @ I5 ) )
                        & ( ( nth_int @ Xsi @ I5 )
                          = ( nth_int @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L6582681566615743372_c_int @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L6582681566615743372_c_int @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_742_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: d > c > assn,A6: d > c > assn,Xs: list_d,Xs4: list_d,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_d @ Xs )
            = ( size_size_list_d @ Xs4 ) )
         => ( ( ( size_size_list_c @ Xsi )
              = ( size_size_list_c @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                   => ( ( ( size_size_list_d @ Xs )
                        = ( size_size_list_c @ Xsi ) )
                     => ( ( ( nth_d @ Xs @ I5 )
                          = ( nth_d @ Xs4 @ I5 ) )
                        & ( ( nth_c @ Xsi @ I5 )
                          = ( nth_c @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7260734717915818975sn_d_c @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L7260734717915818975sn_d_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_743_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: d > d > assn,A6: d > d > assn,Xs: list_d,Xs4: list_d,Xsi: list_d,Xsi2: list_d] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_d @ Xs )
            = ( size_size_list_d @ Xs4 ) )
         => ( ( ( size_size_list_d @ Xsi )
              = ( size_size_list_d @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                   => ( ( ( size_size_list_d @ Xs )
                        = ( size_size_list_d @ Xsi ) )
                     => ( ( ( nth_d @ Xs @ I5 )
                          = ( nth_d @ Xs4 @ I5 ) )
                        & ( ( nth_d @ Xsi @ I5 )
                          = ( nth_d @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7260734717915818976sn_d_d @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L7260734717915818976sn_d_d @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_744_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: d > nat > assn,A6: d > nat > assn,Xs: list_d,Xs4: list_d,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_d @ Xs )
            = ( size_size_list_d @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                   => ( ( ( size_size_list_d @ Xs )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_d @ Xs @ I5 )
                          = ( nth_d @ Xs4 @ I5 ) )
                        & ( ( nth_nat @ Xsi @ I5 )
                          = ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7820616366081188145_d_nat @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L7820616366081188145_d_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_745_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: d > int > assn,A6: d > int > assn,Xs: list_d,Xs4: list_d,Xsi: list_int,Xsi2: list_int] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_d @ Xs )
            = ( size_size_list_d @ Xs4 ) )
         => ( ( ( size_size_list_int @ Xsi )
              = ( size_size_list_int @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
                   => ( ( ( size_size_list_d @ Xs )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_d @ Xs @ I5 )
                          = ( nth_d @ Xs4 @ I5 ) )
                        & ( ( nth_int @ Xsi @ I5 )
                          = ( nth_int @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7818125895572137869_d_int @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L7818125895572137869_d_int @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_746_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: nat > c > assn,A6: nat > c > assn,Xs: list_nat,Xs4: list_nat,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Xs4 ) )
         => ( ( ( size_size_list_c @ Xsi )
              = ( size_size_list_c @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                   => ( ( ( size_size_list_nat @ Xs )
                        = ( size_size_list_c @ Xsi ) )
                     => ( ( ( nth_nat @ Xs @ I5 )
                          = ( nth_nat @ Xs4 @ I5 ) )
                        & ( ( nth_c @ Xsi @ I5 )
                          = ( nth_c @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L8146214880390291086_nat_c @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L8146214880390291086_nat_c @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_747_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: nat > d > assn,A6: nat > d > assn,Xs: list_nat,Xs4: list_nat,Xsi: list_d,Xsi2: list_d] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Xs4 ) )
         => ( ( ( size_size_list_d @ Xsi )
              = ( size_size_list_d @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                   => ( ( ( size_size_list_nat @ Xs )
                        = ( size_size_list_d @ Xsi ) )
                     => ( ( ( nth_nat @ Xs @ I5 )
                          = ( nth_nat @ Xs4 @ I5 ) )
                        & ( ( nth_d @ Xsi @ I5 )
                          = ( nth_d @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L8146214880390291087_nat_d @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L8146214880390291087_nat_d @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_748_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: nat > nat > assn,A6: nat > nat > assn,Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A6 )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
                   => ( ( ( size_size_list_nat @ Xs )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_nat @ Xs @ I5 )
                          = ( nth_nat @ Xs4 @ I5 ) )
                        & ( ( nth_nat @ Xsi @ I5 )
                          = ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7892069108621897922at_nat @ I3 @ A @ Xs @ Xsi )
                = ( vEBT_L7892069108621897922at_nat @ I7 @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_749_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_750_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_751_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_752_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( A2 = B2 )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_753_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( A2 = B2 )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_754_is__singletonE,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_755_is__singletonE,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_756_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o2 @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_757_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X3: nat] :
            ( A
           != ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_758_is__singletonE,axiom,
    ! [A: set_int] :
      ( ( is_singleton_int @ A )
     => ~ ! [X3: int] :
            ( A
           != ( insert_int2 @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_759_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat] :
        ? [X2: produc3843707927480180839at_nat] :
          ( A5
          = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_760_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
        ? [X2: product_prod_nat_nat] :
          ( A5
          = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_761_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X2: $o] :
          ( A5
          = ( insert_o2 @ X2 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_762_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X2: nat] :
          ( A5
          = ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_763_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
        ? [X2: int] :
          ( A5
          = ( insert_int2 @ X2 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_764_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L6585172037124793648_c_nat @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L6585172037124793648_c_nat @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_765_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L6582681566615743372_c_int @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L6582681566615743372_c_int @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_766_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: d > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( vEBT_L7820616366081188145_d_nat @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7820616366081188145_d_nat @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_767_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: d > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( vEBT_L7818125895572137869_d_int @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_d @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7818125895572137869_d_int @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_768_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: nat > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7892069108621897922at_nat @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L7892069108621897922at_nat @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_769_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: nat > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7889578638112847646at_int @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L7889578638112847646at_int @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_770_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: int > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8891422820522952478nt_nat @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8891422820522952478nt_nat @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_771_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: int > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L8888932350013902202nt_int @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8888932350013902202nt_int @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_772_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: c > d > assn,Xsi: list_d] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( vEBT_L824818263033719201sn_c_d @ ( insert_nat2 @ I @ I3 ) @ A @ Xs @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_c @ Xs @ I ) @ ( nth_d @ Xsi @ I ) ) @ ( vEBT_L824818263033719201sn_c_d @ I3 @ A @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_773_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_774_diff__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_775_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_776_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_777_diff__strict__left__mono,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A2 ) @ ( minus_8373710615458151222nteger @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_778_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_779_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A2 @ B2 )
        = ( ord_less_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_780_diff__eq__diff__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A2 @ B2 )
        = ( minus_8373710615458151222nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
        = ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_781_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A2 @ B2 )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_782_bot__empty__eq,axiom,
    ( bot_bo7963750851167320836at_o_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ).

% bot_empty_eq
thf(fact_783_bot__empty__eq,axiom,
    ( bot_bo6829958733157050602_nat_o
    = ( ^ [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq
thf(fact_784_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_785_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_786_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_787_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_788_Collect__empty__eq__bot,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( P = bot_bo7963750851167320836at_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_789_Collect__empty__eq__bot,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( ( collec6321179662152712658at_nat @ P )
        = bot_bo228742789529271731at_nat )
      = ( P = bot_bo6829958733157050602_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_790_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_791_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_792_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_793_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_794_remove__def,axiom,
    ( remove5300788332298244428at_nat
    = ( ^ [X2: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] : ( minus_3314409938677909166at_nat @ A5 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% remove_def
thf(fact_795_remove__def,axiom,
    ( remove6466555014256735590at_nat
    = ( ^ [X2: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% remove_def
thf(fact_796_remove__def,axiom,
    ( remove_o
    = ( ^ [X2: $o,A5: set_o] : ( minus_minus_set_o @ A5 @ ( insert_o2 @ X2 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_797_remove__def,axiom,
    ( remove_int
    = ( ^ [X2: int,A5: set_int] : ( minus_minus_set_int @ A5 @ ( insert_int2 @ X2 @ bot_bot_set_int ) ) ) ) ).

% remove_def
thf(fact_798_remove__def,axiom,
    ( remove_nat
    = ( ^ [X2: nat,A5: set_nat] : ( minus_minus_set_nat @ A5 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_799_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ L ) )
     => ( ( insert8211810215607154385at_nat @ ( nth_Pr7617993195940197384at_nat @ L @ I ) @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ X ) ) )
        = ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_800_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ I @ ( size_s1588839187904960729at_nat @ L ) )
     => ( ( insert9069300056098147895at_nat @ ( nth_Pr1239524528086697198at_nat @ L @ I ) @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ X ) ) )
        = ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_801_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( insert_o2 @ ( nth_o @ L @ I ) @ ( set_o2 @ ( list_update_o @ L @ I @ X ) ) )
        = ( insert_o2 @ X @ ( set_o2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_802_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( insert5290817439147925377n_assn @ ( nth_Pr1769885009046257848n_assn @ L @ I ) @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ X ) ) )
        = ( insert5290817439147925377n_assn @ X @ ( set_Pr1139785259514867910n_assn @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_803_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_c,X: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ L ) )
     => ( ( insert_c @ ( nth_c @ L @ I ) @ ( set_c2 @ ( list_update_c @ L @ I @ X ) ) )
        = ( insert_c @ X @ ( set_c2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_804_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_d,X: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ L ) )
     => ( ( insert_d @ ( nth_d @ L @ I ) @ ( set_d2 @ ( list_update_d @ L @ I @ X ) ) )
        = ( insert_d @ X @ ( set_d2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_805_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( insert_nat2 @ ( nth_nat @ L @ I ) @ ( set_nat2 @ ( list_update_nat @ L @ I @ X ) ) )
        = ( insert_nat2 @ X @ ( set_nat2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_806_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( insert_int2 @ ( nth_int @ L @ I ) @ ( set_int2 @ ( list_update_int @ L @ I @ X ) ) )
        = ( insert_int2 @ X @ ( set_int2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_807_set__swap,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6829681357464350627n_assn @ Xs ) )
       => ( ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ ( list_u4534839942911652127n_assn @ Xs @ I @ ( nth_Pr1769885009046257848n_assn @ Xs @ J ) ) @ J @ ( nth_Pr1769885009046257848n_assn @ Xs @ I ) ) )
          = ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ) ).

% set_swap
thf(fact_808_set__swap,axiom,
    ! [I: nat,Xs: list_c,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_c @ Xs ) )
       => ( ( set_c2 @ ( list_update_c @ ( list_update_c @ Xs @ I @ ( nth_c @ Xs @ J ) ) @ J @ ( nth_c @ Xs @ I ) ) )
          = ( set_c2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_809_set__swap,axiom,
    ! [I: nat,Xs: list_d,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_d @ Xs ) )
       => ( ( set_d2 @ ( list_update_d @ ( list_update_d @ Xs @ I @ ( nth_d @ Xs @ J ) ) @ J @ ( nth_d @ Xs @ I ) ) )
          = ( set_d2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_810_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_811_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_812_Set_Ois__empty__def,axiom,
    ( is_emp7572232645145209145at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat] : A5 = bot_bo228742789529271731at_nat ) ) ).

% Set.is_empty_def
thf(fact_813_Set_Ois__empty__def,axiom,
    ( is_emp1662574758705540307at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] : A5 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
thf(fact_814_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A5: set_o] : A5 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_815_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : A5 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_816_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A5: set_int] : A5 = bot_bot_set_int ) ) ).

% Set.is_empty_def
thf(fact_817_distinct__swap,axiom,
    ! [I: nat,Xs: list_c,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_c @ Xs ) )
       => ( ( distinct_c @ ( list_update_c @ ( list_update_c @ Xs @ I @ ( nth_c @ Xs @ J ) ) @ J @ ( nth_c @ Xs @ I ) ) )
          = ( distinct_c @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_818_distinct__swap,axiom,
    ! [I: nat,Xs: list_d,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_d @ Xs ) )
       => ( ( distinct_d @ ( list_update_d @ ( list_update_d @ Xs @ I @ ( nth_d @ Xs @ J ) ) @ J @ ( nth_d @ Xs @ I ) ) )
          = ( distinct_d @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_819_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_820_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( distinct_int @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_821_list__ex__length,axiom,
    ( list_ex_c
    = ( ^ [P2: c > $o,Xs3: list_c] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs3 ) )
          & ( P2 @ ( nth_c @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_822_list__ex__length,axiom,
    ( list_ex_d
    = ( ^ [P2: d > $o,Xs3: list_d] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs3 ) )
          & ( P2 @ ( nth_d @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_823_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P2: nat > $o,Xs3: list_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs3 ) )
          & ( P2 @ ( nth_nat @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_824_list__ex__length,axiom,
    ( list_ex_int
    = ( ^ [P2: int > $o,Xs3: list_int] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs3 ) )
          & ( P2 @ ( nth_int @ Xs3 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_825_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B2 ) ) )
      = ( ! [H3: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H3 )
           => B2 )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_826_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_827_member__remove,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( remove5300788332298244428at_nat @ Y @ A ) )
      = ( ( member8757157785044589968at_nat @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_828_member__remove,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( remove6466555014256735590at_nat @ Y @ A ) )
      = ( ( member8440522571783428010at_nat @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_829_member__remove,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
      = ( ( member_nat @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_830_member__remove,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( member_int @ X @ ( remove_int @ Y @ A ) )
      = ( ( member_int @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_831_member__remove,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( remove4651630035290841522_nat_o @ Y @ A ) )
      = ( ( member6576561426505652726_nat_o @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_832_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      ( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_star
thf(fact_833_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_834_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_835_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B2 ) ) ).

% mod_pure_star_dist
thf(fact_836_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ Q )
      = ( B2
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_837_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_838_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B2 ) ).

% mod_h_bot_iff(1)
thf(fact_839_list__ex__cong,axiom,
    ! [Xs: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat,F4: produc3843707927480180839at_nat > $o,G: produc3843707927480180839at_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e5181628221095382443at_nat @ F4 @ Xs )
          = ( list_e5181628221095382443at_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_840_list__ex__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F4: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e7689525607045846085at_nat @ F4 @ Xs )
          = ( list_e7689525607045846085at_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_841_list__ex__cong,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,Ys: list_P7985473006766602707_nat_o,F4: ( produc3658429121746597890et_nat > $o ) > $o,G: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ ( set_Pr400311997697478754_nat_o @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e4587384546317177617_nat_o @ F4 @ Xs )
          = ( list_e4587384546317177617_nat_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_842_list__ex__cong,axiom,
    ! [Xs: list_int,Ys: list_int,F4: int > $o,G: int > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_int @ F4 @ Xs )
          = ( list_ex_int @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_843_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F4: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_nat @ F4 @ Xs )
          = ( list_ex_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_844_list__ex__cong,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,F4: produc6575502325842934193n_assn > $o,G: produc6575502325842934193n_assn > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Ys ) )
           => ( ( F4 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e1087220904399170805n_assn @ F4 @ Xs )
          = ( list_e1087220904399170805n_assn @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_845_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_846_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_847_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H @ As ) ) ).

% one_assn_raw.cases
thf(fact_848_bex2I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ S )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_849_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_850_bex2I,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
     => ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_851_bex2I,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_852_bex2I,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_853_bex2I,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_854_bex2I,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_855_pairself_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F5: int > option6357759511663192854e_term,A4: int,B4: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F5 @ ( product_Pair_int_int @ A4 @ B4 ) ) ) ).

% pairself.cases
thf(fact_856_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A: int > int > assn,A6: int > int > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi: int] :
              ( ( member_int @ X3 @ ( set_int2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L74593716426352029nt_int @ A @ Xs @ Xsi )
            = ( vEBT_L74593716426352029nt_int @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_857_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_nat,Xsi2: list_nat,A: int > nat > assn,A6: int > nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi: nat] :
              ( ( member_int @ X3 @ ( set_int2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L77084186935402305nt_nat @ A @ Xs @ Xsi )
            = ( vEBT_L77084186935402305nt_nat @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_858_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_int,Xsi2: list_int,A: nat > int > assn,A6: nat > int > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: int] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8298612041380073281at_int @ A @ Xs @ Xsi )
            = ( vEBT_L8298612041380073281at_int @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_859_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A: nat > nat > assn,A6: nat > nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8301102511889123557at_nat @ A @ Xs @ Xsi )
            = ( vEBT_L8301102511889123557at_nat @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_860_list__assn__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Xsi: list_int,Xsi2: list_int,A: product_prod_nat_nat > int > assn,A6: product_prod_nat_nat > int > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: product_prod_nat_nat,Xi: int] :
              ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L6295016686126367274at_int @ A @ Xs @ Xsi )
            = ( vEBT_L6295016686126367274at_int @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_861_list__assn__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Xsi: list_nat,Xsi2: list_nat,A: product_prod_nat_nat > nat > assn,A6: product_prod_nat_nat > nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: product_prod_nat_nat,Xi: nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L6297507156635417550at_nat @ A @ Xs @ Xsi )
            = ( vEBT_L6297507156635417550at_nat @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_862_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_P6011104703257516679at_nat,Xsi2: list_P6011104703257516679at_nat,A: int > product_prod_nat_nat > assn,A6: int > product_prod_nat_nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi: product_prod_nat_nat] :
              ( ( member_int @ X3 @ ( set_int2 @ Xs4 ) )
             => ( ( member8440522571783428010at_nat @ Xi @ ( set_Pr5648618587558075414at_nat @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L3472024597287490640at_nat @ A @ Xs @ Xsi )
            = ( vEBT_L3472024597287490640at_nat @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_863_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_P8527749157015355191n_assn,Xsi2: list_P8527749157015355191n_assn,A: int > produc6575502325842934193n_assn > assn,A6: int > produc6575502325842934193n_assn > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi: produc6575502325842934193n_assn] :
              ( ( member_int @ X3 @ ( set_int2 @ Xs4 ) )
             => ( ( member7957490590177025114n_assn @ Xi @ ( set_Pr1139785259514867910n_assn @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L1620274777744359680n_assn @ A @ Xs @ Xsi )
            = ( vEBT_L1620274777744359680n_assn @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_864_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_P6011104703257516679at_nat,Xsi2: list_P6011104703257516679at_nat,A: nat > product_prod_nat_nat > assn,A6: nat > product_prod_nat_nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: product_prod_nat_nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member8440522571783428010at_nat @ Xi @ ( set_Pr5648618587558075414at_nat @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L434182512823356844at_nat @ A @ Xs @ Xsi )
            = ( vEBT_L434182512823356844at_nat @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_865_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_P8527749157015355191n_assn,Xsi2: list_P8527749157015355191n_assn,A: nat > produc6575502325842934193n_assn > assn,A6: nat > produc6575502325842934193n_assn > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: nat,Xi: produc6575502325842934193n_assn] :
              ( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
             => ( ( member7957490590177025114n_assn @ Xi @ ( set_Pr1139785259514867910n_assn @ Xsi2 ) )
               => ( ( A @ X3 @ Xi )
                  = ( A6 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L9083610150846024284n_assn @ A @ Xs @ Xsi )
            = ( vEBT_L9083610150846024284n_assn @ A6 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_866_distinct__Ex1,axiom,
    ! [Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( ( distin5355857245021852384at_nat @ Xs )
     => ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s1588839187904960729at_nat @ Xs ) )
            & ( ( nth_Pr1239524528086697198at_nat @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_s1588839187904960729at_nat @ Xs ) )
                  & ( ( nth_Pr1239524528086697198at_nat @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_867_distinct__Ex1,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_s5460976970255530739at_nat @ Xs ) )
                  & ( ( nth_Pr7617993195940197384at_nat @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_868_distinct__Ex1,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( distin299582740263731014_nat_o @ Xs )
     => ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s8908918004648414271_nat_o @ Xs ) )
            & ( ( nth_Pr6840583978799713108_nat_o @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_s8908918004648414271_nat_o @ Xs ) )
                  & ( ( nth_Pr6840583978799713108_nat_o @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_869_distinct__Ex1,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( distin4809818325527169450n_assn @ Xs )
     => ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s6829681357464350627n_assn @ Xs ) )
            & ( ( nth_Pr1769885009046257848n_assn @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_s6829681357464350627n_assn @ Xs ) )
                  & ( ( nth_Pr1769885009046257848n_assn @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_870_distinct__Ex1,axiom,
    ! [Xs: list_c,X: c] :
      ( ( distinct_c @ Xs )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_c @ Xs ) )
            & ( ( nth_c @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_size_list_c @ Xs ) )
                  & ( ( nth_c @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_871_distinct__Ex1,axiom,
    ! [Xs: list_d,X: d] :
      ( ( distinct_d @ Xs )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_d @ Xs ) )
            & ( ( nth_d @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_size_list_d @ Xs ) )
                  & ( ( nth_d @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_872_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_873_distinct__Ex1,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ X3 )
              = X )
            & ! [Y3: nat] :
                ( ( ( ord_less_nat @ Y3 @ ( size_size_list_int @ Xs ) )
                  & ( ( nth_int @ Xs @ Y3 )
                    = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_874_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_875_distinct__list__update,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,A2: produc3658429121746597890et_nat > $o,I: nat] :
      ( ( distin299582740263731014_nat_o @ Xs )
     => ( ~ ( member6576561426505652726_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ ( set_Pr400311997697478754_nat_o @ Xs ) @ ( insert5175938949040314269_nat_o @ ( nth_Pr6840583978799713108_nat_o @ Xs @ I ) @ bot_bo7824918357723582233_nat_o ) ) )
       => ( distin299582740263731014_nat_o @ ( list_u6943956310655620667_nat_o @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_876_distinct__list__update,axiom,
    ! [Xs: list_P8527749157015355191n_assn,A2: produc6575502325842934193n_assn,I: nat] :
      ( ( distin4809818325527169450n_assn @ Xs )
     => ( ~ ( member7957490590177025114n_assn @ A2 @ ( minus_2301194108115264760n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( insert5290817439147925377n_assn @ ( nth_Pr1769885009046257848n_assn @ Xs @ I ) @ bot_bo1176836662018730877n_assn ) ) )
       => ( distin4809818325527169450n_assn @ ( list_u4534839942911652127n_assn @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_877_distinct__list__update,axiom,
    ! [Xs: list_c,A2: c,I: nat] :
      ( ( distinct_c @ Xs )
     => ( ~ ( member_c @ A2 @ ( minus_minus_set_c @ ( set_c2 @ Xs ) @ ( insert_c @ ( nth_c @ Xs @ I ) @ bot_bot_set_c ) ) )
       => ( distinct_c @ ( list_update_c @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_878_distinct__list__update,axiom,
    ! [Xs: list_d,A2: d,I: nat] :
      ( ( distinct_d @ Xs )
     => ( ~ ( member_d @ A2 @ ( minus_minus_set_d @ ( set_d2 @ Xs ) @ ( insert_d @ ( nth_d @ Xs @ I ) @ bot_bot_set_d ) ) )
       => ( distinct_d @ ( list_update_d @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_879_distinct__list__update,axiom,
    ! [Xs: list_P5464809261938338413at_nat,A2: produc3843707927480180839at_nat,I: nat] :
      ( ( distin5355857245021852384at_nat @ Xs )
     => ( ~ ( member8757157785044589968at_nat @ A2 @ ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ ( nth_Pr1239524528086697198at_nat @ Xs @ I ) @ bot_bo228742789529271731at_nat ) ) )
       => ( distin5355857245021852384at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_880_distinct__list__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A2: product_prod_nat_nat,I: nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ~ ( member8440522571783428010at_nat @ A2 @ ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ bot_bo2099793752762293965at_nat ) ) )
       => ( distin6923225563576452346at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_881_distinct__list__update,axiom,
    ! [Xs: list_o,A2: $o,I: nat] :
      ( ( distinct_o @ Xs )
     => ( ~ ( member_o @ A2 @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ ( nth_o @ Xs @ I ) @ bot_bot_set_o ) ) )
       => ( distinct_o @ ( list_update_o @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_882_distinct__list__update,axiom,
    ! [Xs: list_int,A2: int,I: nat] :
      ( ( distinct_int @ Xs )
     => ( ~ ( member_int @ A2 @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ ( nth_int @ Xs @ I ) @ bot_bot_set_int ) ) )
       => ( distinct_int @ ( list_update_int @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_883_distinct__list__update,axiom,
    ! [Xs: list_nat,A2: nat,I: nat] :
      ( ( distinct_nat @ Xs )
     => ( ~ ( member_nat @ A2 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
       => ( distinct_nat @ ( list_update_nat @ Xs @ I @ A2 ) ) ) ) ).

% distinct_list_update
thf(fact_884_distinct__conv__nth,axiom,
    ( distinct_c
    = ( ^ [Xs3: list_c] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_c @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_c @ Xs3 ) )
             => ( ( I4 != J2 )
               => ( ( nth_c @ Xs3 @ I4 )
                 != ( nth_c @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_885_distinct__conv__nth,axiom,
    ( distinct_d
    = ( ^ [Xs3: list_d] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_d @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_d @ Xs3 ) )
             => ( ( I4 != J2 )
               => ( ( nth_d @ Xs3 @ I4 )
                 != ( nth_d @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_886_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs3: list_nat] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( I4 != J2 )
               => ( ( nth_nat @ Xs3 @ I4 )
                 != ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_887_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs3: list_int] :
        ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs3 ) )
             => ( ( I4 != J2 )
               => ( ( nth_int @ Xs3 @ I4 )
                 != ( nth_int @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_888_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_c,I: nat,J: nat] :
      ( ( distinct_c @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_c @ Xs ) )
         => ( ( ( nth_c @ Xs @ I )
              = ( nth_c @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_889_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_d,I: nat,J: nat] :
      ( ( distinct_d @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_d @ Xs ) )
         => ( ( ( nth_d @ Xs @ I )
              = ( nth_d @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_890_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_891_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ( ( nth_int @ Xs @ I )
              = ( nth_int @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_892_all__set__conv__nth,axiom,
    ! [L: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X2: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ L ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ L @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_893_all__set__conv__nth,axiom,
    ! [L: list_c,P: c > $o] :
      ( ( ! [X2: c] :
            ( ( member_c @ X2 @ ( set_c2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_c @ L ) )
           => ( P @ ( nth_c @ L @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_894_all__set__conv__nth,axiom,
    ! [L: list_d,P: d > $o] :
      ( ( ! [X2: d] :
            ( ( member_d @ X2 @ ( set_d2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_d @ L ) )
           => ( P @ ( nth_d @ L @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_895_all__set__conv__nth,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ L ) )
           => ( P @ ( nth_nat @ L @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_896_all__set__conv__nth,axiom,
    ! [L: list_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ L ) )
           => ( P @ ( nth_int @ L @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_897_nth__mem,axiom,
    ! [N: nat,Xs: list_P5464809261938338413at_nat] :
      ( ( ord_less_nat @ N @ ( size_s1588839187904960729at_nat @ Xs ) )
     => ( member8757157785044589968at_nat @ ( nth_Pr1239524528086697198at_nat @ Xs @ N ) @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_898_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_899_nth__mem,axiom,
    ! [N: nat,Xs: list_P7985473006766602707_nat_o] :
      ( ( ord_less_nat @ N @ ( size_s8908918004648414271_nat_o @ Xs ) )
     => ( member6576561426505652726_nat_o @ ( nth_Pr6840583978799713108_nat_o @ Xs @ N ) @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% nth_mem
thf(fact_900_nth__mem,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( member7957490590177025114n_assn @ ( nth_Pr1769885009046257848n_assn @ Xs @ N ) @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ).

% nth_mem
thf(fact_901_nth__mem,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
     => ( member_c @ ( nth_c @ Xs @ N ) @ ( set_c2 @ Xs ) ) ) ).

% nth_mem
thf(fact_902_nth__mem,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
     => ( member_d @ ( nth_d @ Xs @ N ) @ ( set_d2 @ Xs ) ) ) ).

% nth_mem
thf(fact_903_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_904_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_905_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_906_list__ball__nth,axiom,
    ! [N: nat,Xs: list_c,P: c > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
     => ( ! [X3: c] :
            ( ( member_c @ X3 @ ( set_c2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_c @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_907_list__ball__nth,axiom,
    ! [N: nat,Xs: list_d,P: d > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
     => ( ! [X3: d] :
            ( ( member_d @ X3 @ ( set_d2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_d @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_908_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_909_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_910_in__set__conv__nth,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s1588839187904960729at_nat @ Xs ) )
            & ( ( nth_Pr1239524528086697198at_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_911_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_912_in__set__conv__nth,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8908918004648414271_nat_o @ Xs ) )
            & ( ( nth_Pr6840583978799713108_nat_o @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_913_in__set__conv__nth,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ Xs ) )
            & ( ( nth_Pr1769885009046257848n_assn @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_914_in__set__conv__nth,axiom,
    ! [X: c,Xs: list_c] :
      ( ( member_c @ X @ ( set_c2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_c @ Xs ) )
            & ( ( nth_c @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_915_in__set__conv__nth,axiom,
    ! [X: d,Xs: list_d] :
      ( ( member_d @ X @ ( set_d2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_d @ Xs ) )
            & ( ( nth_d @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_916_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_917_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_918_all__nth__imp__all__set,axiom,
    ! [Xs: list_P5464809261938338413at_nat,P: produc3843707927480180839at_nat > $o,X: produc3843707927480180839at_nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s1588839187904960729at_nat @ Xs ) )
         => ( P @ ( nth_Pr1239524528086697198at_nat @ Xs @ I5 ) ) )
     => ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_919_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s5460976970255530739at_nat @ Xs ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I5 ) ) )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_920_all__nth__imp__all__set,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s8908918004648414271_nat_o @ Xs ) )
         => ( P @ ( nth_Pr6840583978799713108_nat_o @ Xs @ I5 ) ) )
     => ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_921_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s6829681357464350627n_assn @ Xs ) )
         => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs @ I5 ) ) )
     => ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_922_all__nth__imp__all__set,axiom,
    ! [Xs: list_c,P: c > $o,X: c] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_c @ Xs ) )
         => ( P @ ( nth_c @ Xs @ I5 ) ) )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_923_all__nth__imp__all__set,axiom,
    ! [Xs: list_d,P: d > $o,X: d] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_d @ Xs ) )
         => ( P @ ( nth_d @ Xs @ I5 ) ) )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_924_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I5 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_925_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I5 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_926_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X2: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ Xs ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_927_all__set__conv__all__nth,axiom,
    ! [Xs: list_c,P: c > $o] :
      ( ( ! [X2: c] :
            ( ( member_c @ X2 @ ( set_c2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_c @ Xs ) )
           => ( P @ ( nth_c @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_928_all__set__conv__all__nth,axiom,
    ! [Xs: list_d,P: d > $o] :
      ( ( ! [X2: d] :
            ( ( member_d @ X2 @ ( set_d2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_d @ Xs ) )
           => ( P @ ( nth_d @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_929_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_930_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_931_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ I @ ( size_s1588839187904960729at_nat @ L ) )
     => ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_932_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ L ) )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_933_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P7985473006766602707_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( ord_less_nat @ I @ ( size_s8908918004648414271_nat_o @ L ) )
     => ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_934_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_935_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_c,X: c,Y: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ L ) )
     => ( ( member_c @ X @ ( set_c2 @ ( list_update_c @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: c] : ( member_c @ X @ ( set_c2 @ ( list_update_c @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_936_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_d,X: d,Y: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ L ) )
     => ( ( member_d @ X @ ( set_d2 @ ( list_update_d @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: d] : ( member_d @ X @ ( set_d2 @ ( list_update_d @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_937_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_938_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_939_in__set__upd__cases,axiom,
    ! [X: produc3843707927480180839at_nat,L: list_P5464809261938338413at_nat,I: nat,Y: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s1588839187904960729at_nat @ L ) )
         => ( X != Y ) )
       => ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_940_in__set__upd__cases,axiom,
    ! [X: product_prod_nat_nat,L: list_P6011104703257516679at_nat,I: nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ L ) )
         => ( X != Y ) )
       => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_941_in__set__upd__cases,axiom,
    ! [X: produc3658429121746597890et_nat > $o,L: list_P7985473006766602707_nat_o,I: nat,Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s8908918004648414271_nat_o @ L ) )
         => ( X != Y ) )
       => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_942_in__set__upd__cases,axiom,
    ! [X: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
         => ( X != Y ) )
       => ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_943_in__set__upd__cases,axiom,
    ! [X: c,L: list_c,I: nat,Y: c] :
      ( ( member_c @ X @ ( set_c2 @ ( list_update_c @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_c @ L ) )
         => ( X != Y ) )
       => ( member_c @ X @ ( set_c2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_944_in__set__upd__cases,axiom,
    ! [X: d,L: list_d,I: nat,Y: d] :
      ( ( member_d @ X @ ( set_d2 @ ( list_update_d @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_d @ L ) )
         => ( X != Y ) )
       => ( member_d @ X @ ( set_d2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_945_in__set__upd__cases,axiom,
    ! [X: nat,L: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( X != Y ) )
       => ( member_nat @ X @ ( set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_946_in__set__upd__cases,axiom,
    ! [X: int,L: list_int,I: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( X != Y ) )
       => ( member_int @ X @ ( set_int2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_947_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ I @ ( size_s1588839187904960729at_nat @ L ) )
     => ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ L ) )
            & ! [Y5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_948_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ L ) )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ L ) )
            & ! [Y5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_949_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P7985473006766602707_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( ord_less_nat @ I @ ( size_s8908918004648414271_nat_o @ L ) )
     => ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ L ) )
            & ! [Y5: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_950_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L ) )
     => ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ L ) )
            & ! [Y5: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_951_in__set__upd__eq,axiom,
    ! [I: nat,L: list_c,X: c,Y: c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ L ) )
     => ( ( member_c @ X @ ( set_c2 @ ( list_update_c @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_c @ X @ ( set_c2 @ L ) )
            & ! [Y5: c] : ( member_c @ X @ ( set_c2 @ ( list_update_c @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_952_in__set__upd__eq,axiom,
    ! [I: nat,L: list_d,X: d,Y: d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ L ) )
     => ( ( member_d @ X @ ( set_d2 @ ( list_update_d @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_d @ X @ ( set_d2 @ L ) )
            & ! [Y5: d] : ( member_d @ X @ ( set_d2 @ ( list_update_d @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_953_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_nat @ X @ ( set_nat2 @ L ) )
            & ! [Y5: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_954_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_int @ X @ ( set_int2 @ L ) )
            & ! [Y5: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_955_set__update__memI,axiom,
    ! [N: nat,Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ N @ ( size_s1588839187904960729at_nat @ Xs ) )
     => ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_956_set__update__memI,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_957_set__update__memI,axiom,
    ! [N: nat,Xs: list_P7985473006766602707_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s8908918004648414271_nat_o @ Xs ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_958_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_959_set__update__memI,axiom,
    ! [N: nat,Xs: list_c,X: c] :
      ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
     => ( member_c @ X @ ( set_c2 @ ( list_update_c @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_960_set__update__memI,axiom,
    ! [N: nat,Xs: list_d,X: d] :
      ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
     => ( member_d @ X @ ( set_d2 @ ( list_update_d @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_961_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_962_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_963_set__update__distinct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat,X: produc6575502325842934193n_assn] :
      ( ( distin4809818325527169450n_assn @ Xs )
     => ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
       => ( ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs @ N @ X ) )
          = ( insert5290817439147925377n_assn @ X @ ( minus_2301194108115264760n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( insert5290817439147925377n_assn @ ( nth_Pr1769885009046257848n_assn @ Xs @ N ) @ bot_bo1176836662018730877n_assn ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_964_set__update__distinct,axiom,
    ! [Xs: list_c,N: nat,X: c] :
      ( ( distinct_c @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
       => ( ( set_c2 @ ( list_update_c @ Xs @ N @ X ) )
          = ( insert_c @ X @ ( minus_minus_set_c @ ( set_c2 @ Xs ) @ ( insert_c @ ( nth_c @ Xs @ N ) @ bot_bot_set_c ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_965_set__update__distinct,axiom,
    ! [Xs: list_d,N: nat,X: d] :
      ( ( distinct_d @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
       => ( ( set_d2 @ ( list_update_d @ Xs @ N @ X ) )
          = ( insert_d @ X @ ( minus_minus_set_d @ ( set_d2 @ Xs ) @ ( insert_d @ ( nth_d @ Xs @ N ) @ bot_bot_set_d ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_966_set__update__distinct,axiom,
    ! [Xs: list_P5464809261938338413at_nat,N: nat,X: produc3843707927480180839at_nat] :
      ( ( distin5355857245021852384at_nat @ Xs )
     => ( ( ord_less_nat @ N @ ( size_s1588839187904960729at_nat @ Xs ) )
       => ( ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ N @ X ) )
          = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ ( nth_Pr1239524528086697198at_nat @ Xs @ N ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_967_set__update__distinct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
       => ( ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) )
          = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_968_set__update__distinct,axiom,
    ! [Xs: list_o,N: nat,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ Xs @ N @ X ) )
          = ( insert_o2 @ X @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ ( nth_o @ Xs @ N ) @ bot_bot_set_o ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_969_set__update__distinct,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ Xs @ N @ X ) )
          = ( insert_int2 @ X @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ ( nth_int @ Xs @ N ) @ bot_bot_set_int ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_970_set__update__distinct,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
          = ( insert_nat2 @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_971_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
        = ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_972_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_973_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_974_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_975_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_976_old_Oprod_Oinject,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A2 @ B2 )
        = ( produc8603105652947943368nteger @ A7 @ B6 ) )
      = ( ( A2 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_977_old_Oprod_Oinject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,A7: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A2 @ B2 )
        = ( produc5001842942810119800et_nat @ A7 @ B6 ) )
      = ( ( A2 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_978_old_Oprod_Oinject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,A7: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A2 @ B2 )
        = ( produc2245416461498447860et_nat @ A7 @ B6 ) )
      = ( ( A2 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_979_old_Oprod_Oinject,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A2 @ B2 )
        = ( produc5700946648718959541nt_int @ A7 @ B6 ) )
      = ( ( A2 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_980_old_Oprod_Oinject,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,A7: int > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A2 @ B2 )
        = ( produc4305682042979456191nt_int @ A7 @ B6 ) )
      = ( ( A2 = A7 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_981_distinct__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( distinct_int @ ( union_int @ Xs @ Ys ) )
      = ( distinct_int @ Ys ) ) ).

% distinct_union
thf(fact_982_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_983_set__remove1__eq,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( distin4809818325527169450n_assn @ Xs )
     => ( ( set_Pr1139785259514867910n_assn @ ( remove1670527618125605709n_assn @ X @ Xs ) )
        = ( minus_2301194108115264760n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( insert5290817439147925377n_assn @ X @ bot_bo1176836662018730877n_assn ) ) ) ) ).

% set_remove1_eq
thf(fact_984_set__remove1__eq,axiom,
    ! [Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( ( distin5355857245021852384at_nat @ Xs )
     => ( ( set_Pr3765526544606949372at_nat @ ( remove8895909992688781315at_nat @ X @ Xs ) )
        = ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_985_set__remove1__eq,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( set_Pr5648618587558075414at_nat @ ( remove5766854097993331357at_nat @ X @ Xs ) )
        = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_986_set__remove1__eq,axiom,
    ! [Xs: list_o,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( set_o2 @ ( remove1_o @ X @ Xs ) )
        = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% set_remove1_eq
thf(fact_987_set__remove1__eq,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( set_int2 @ ( remove1_int @ X @ Xs ) )
        = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).

% set_remove1_eq
thf(fact_988_set__remove1__eq,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( set_nat2 @ ( remove1_nat @ X @ Xs ) )
        = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_989_is__empty__set,axiom,
    ! [Xs: list_int] :
      ( ( is_empty_int @ ( set_int2 @ Xs ) )
      = ( null_int @ Xs ) ) ).

% is_empty_set
thf(fact_990_is__empty__set,axiom,
    ! [Xs: list_nat] :
      ( ( is_empty_nat @ ( set_nat2 @ Xs ) )
      = ( null_nat @ Xs ) ) ).

% is_empty_set
thf(fact_991_is__empty__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( is_emp7749065762491207043n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) )
      = ( null_P7596217980778516671n_assn @ Xs ) ) ).

% is_empty_set
thf(fact_992_set__removeAll,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( remove1763081955827946941n_assn @ X @ Xs ) )
      = ( minus_2301194108115264760n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( insert5290817439147925377n_assn @ X @ bot_bo1176836662018730877n_assn ) ) ) ).

% set_removeAll
thf(fact_993_set__removeAll,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( remove4908499291543849587at_nat @ X @ Xs ) )
      = ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).

% set_removeAll
thf(fact_994_set__removeAll,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) )
      = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_removeAll
thf(fact_995_set__removeAll,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( set_o2 @ ( removeAll_o @ X @ Xs ) )
      = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ).

% set_removeAll
thf(fact_996_set__removeAll,axiom,
    ! [X: int,Xs: list_int] :
      ( ( set_int2 @ ( removeAll_int @ X @ Xs ) )
      = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ).

% set_removeAll
thf(fact_997_set__removeAll,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).

% set_removeAll
thf(fact_998_listrel1__iff__update,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( listre8180072825634266816at_nat @ R2 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat,N2: nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s8736152011456118867at_nat @ Xs ) )
            & ( Ys
              = ( list_u8444657558853818831at_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_999_listrel1__iff__update,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( listrel1_c @ R2 ) )
      = ( ? [Y5: c,N2: nat] :
            ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ ( nth_c @ Xs @ N2 ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
            & ( Ys
              = ( list_update_c @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1000_listrel1__iff__update,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( listrel1_d @ R2 ) )
      = ( ? [Y5: d,N2: nat] :
            ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ ( nth_d @ Xs @ N2 ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs ) )
            & ( Ys
              = ( list_update_d @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1001_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y5: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1002_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
      = ( ? [Y5: int,N2: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ Y5 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1003_List_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( insert8944034826898310173at_nat @ X @ Xs ) )
      = ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% List.set_insert
thf(fact_1004_List_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( insert4149901806602606979at_nat @ X @ Xs ) )
      = ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% List.set_insert
thf(fact_1005_List_Oset__insert,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( set_o2 @ ( insert_o @ X @ Xs ) )
      = ( insert_o2 @ X @ ( set_o2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_1006_List_Oset__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( set_int2 @ ( insert_int @ X @ Xs ) )
      = ( insert_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_1007_List_Oset__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
      = ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_1008_List_Oset__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( insert3246601298802261197n_assn @ X @ Xs ) )
      = ( insert5290817439147925377n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ).

% List.set_insert
thf(fact_1009_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( listrel_c_c @ R2 ) )
      = ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_c @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
           => ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ ( nth_c @ Xs @ N2 ) @ ( nth_c @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1010_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_d,R2: set_Product_prod_c_d] :
      ( ( member7083281593699519441list_d @ ( produc5515445822533511192list_d @ Xs @ Ys ) @ ( listrel_c_d @ R2 ) )
      = ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_d @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
           => ( member5074992363344545361od_c_d @ ( product_Pair_c_d @ ( nth_c @ Xs @ N2 ) @ ( nth_d @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1011_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_nat,R2: set_Pr6903500605879609269_c_nat] :
      ( ( member6184937725913850668st_nat @ ( produc6126748735280710287st_nat @ Xs @ Ys ) @ ( listrel_c_nat @ R2 ) )
      = ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
           => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ ( nth_c @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1012_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_int,R2: set_Pr4413417460947882513_c_int] :
      ( ( member3694854580982123912st_int @ ( produc1948897715771513579st_int @ Xs @ Ys ) @ ( listrel_c_int @ R2 ) )
      = ( ( ( size_size_list_c @ Xs )
          = ( size_size_list_int @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
           => ( member4017226226790010994_c_int @ ( product_Pair_c_int @ ( nth_c @ Xs @ N2 ) @ ( nth_int @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1013_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_c,R2: set_Product_prod_d_c] :
      ( ( member1739767152416986127list_c @ ( produc242965416994095062list_c @ Xs @ Ys ) @ ( listrel_d_c @ R2 ) )
      = ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_c @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs ) )
           => ( member2287536777068640527od_d_c @ ( product_Pair_d_c @ ( nth_d @ Xs @ N2 ) @ ( nth_c @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1014_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( listrel_d_d @ R2 ) )
      = ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_d @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs ) )
           => ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ ( nth_d @ Xs @ N2 ) @ ( nth_d @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1015_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_nat,R2: set_Pr3276347184212964150_d_nat] :
      ( ( member6851837201453759021st_nat @ ( produc6793648210820618640st_nat @ Xs @ Ys ) @ ( listrel_d_nat @ R2 ) )
      = ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs ) )
           => ( member207149538400826391_d_nat @ ( product_Pair_d_nat @ ( nth_d @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1016_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_int,R2: set_Pr786264039281237394_d_int] :
      ( ( member4361754056522032265st_int @ ( produc2615797191311421932st_int @ Xs @ Ys ) @ ( listrel_d_int @ R2 ) )
      = ( ( ( size_size_list_d @ Xs )
          = ( size_size_list_int @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_d @ Xs ) )
           => ( member5252670555746405491_d_int @ ( product_Pair_d_int @ ( nth_d @ Xs @ N2 ) @ ( nth_int @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1017_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_c,R2: set_Pr4335409928928841947_nat_c] :
      ( ( member6074218473364765234list_c @ ( produc7723716018658481613list_c @ Xs @ Ys ) @ ( listrel_nat_c @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_c @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8962352060716553276_nat_c @ ( product_Pair_nat_c @ ( nth_nat @ Xs @ N2 ) @ ( nth_c @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1018_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_d,R2: set_Pr4406443968975187932_nat_d] :
      ( ( member6145252513411111219list_d @ ( produc7723716022961710414list_d @ Xs @ Ys ) @ ( listrel_nat_d @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_d @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8962352065019782077_nat_d @ ( product_Pair_nat_d @ ( nth_nat @ Xs @ N2 ) @ ( nth_d @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1019_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( ( entails @ P @ ( pure_assn @ B2 ) )
      = ( ! [H3: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H3 )
           => B2 )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_1020_mult__1,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1021_mult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1022_mult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1023_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1024_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_1025_mult_Oright__neutral,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% mult.right_neutral
thf(fact_1026_mult_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.right_neutral
thf(fact_1027_mult_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.right_neutral
thf(fact_1028_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_1029_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_1030_in__set__remove1,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( A2 != B2 )
     => ( ( member8757157785044589968at_nat @ A2 @ ( set_Pr3765526544606949372at_nat @ ( remove8895909992688781315at_nat @ B2 @ Xs ) ) )
        = ( member8757157785044589968at_nat @ A2 @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1031_in__set__remove1,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( A2 != B2 )
     => ( ( member8440522571783428010at_nat @ A2 @ ( set_Pr5648618587558075414at_nat @ ( remove5766854097993331357at_nat @ B2 @ Xs ) ) )
        = ( member8440522571783428010at_nat @ A2 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1032_in__set__remove1,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( A2 != B2 )
     => ( ( member6576561426505652726_nat_o @ A2 @ ( set_Pr400311997697478754_nat_o @ ( remove4235003712227595625_nat_o @ B2 @ Xs ) ) )
        = ( member6576561426505652726_nat_o @ A2 @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1033_in__set__remove1,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( ( A2 != B2 )
     => ( ( member_int @ A2 @ ( set_int2 @ ( remove1_int @ B2 @ Xs ) ) )
        = ( member_int @ A2 @ ( set_int2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1034_in__set__remove1,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( A2 != B2 )
     => ( ( member_nat @ A2 @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs ) ) )
        = ( member_nat @ A2 @ ( set_nat2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1035_in__set__remove1,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( A2 != B2 )
     => ( ( member7957490590177025114n_assn @ A2 @ ( set_Pr1139785259514867910n_assn @ ( remove1670527618125605709n_assn @ B2 @ Xs ) ) )
        = ( member7957490590177025114n_assn @ A2 @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1036_removeAll__id,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( remove4908499291543849587at_nat @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1037_removeAll__id,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( remove3673390508374433037at_nat @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1038_removeAll__id,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( remove7604696131425036249_nat_o @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1039_removeAll__id,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( removeAll_int @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1040_removeAll__id,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( removeAll_nat @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1041_removeAll__id,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( remove1763081955827946941n_assn @ X @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1042_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_1043_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_1044_entt__emp,axiom,
    ! [A: assn] : ( entailst @ A @ one_one_assn ) ).

% entt_emp
thf(fact_1045_in__set__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( insert4149901806602606979at_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1046_in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1047_in__set__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( insert8021100735160035049_nat_o @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1048_in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( insert_int @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1049_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1050_in__set__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( insert3246601298802261197n_assn @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1051_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_1052_distinct__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( distinct_int @ ( insert_int @ X @ Xs ) )
      = ( distinct_int @ Xs ) ) ).

% distinct_insert
thf(fact_1053_distinct__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( insert_nat @ X @ Xs ) )
      = ( distinct_nat @ Xs ) ) ).

% distinct_insert
thf(fact_1054_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B2 ) @ Q )
      = ( B2
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_1055_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_1056_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_1057_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_1058_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_1059_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_1060_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P5: produc3658429121746597890et_nat > $o,Q4: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P5 @ ( produc5001842942810119800et_nat @ Q4 @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_1061_listrel1__eq__len,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( listrel1_c @ R2 ) )
     => ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1062_listrel1__eq__len,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( listrel1_d @ R2 ) )
     => ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1063_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1064_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1065_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( listrel_c_c @ R2 ) )
     => ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1066_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_d,R2: set_Product_prod_c_d] :
      ( ( member7083281593699519441list_d @ ( produc5515445822533511192list_d @ Xs @ Ys ) @ ( listrel_c_d @ R2 ) )
     => ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1067_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_nat,R2: set_Pr6903500605879609269_c_nat] :
      ( ( member6184937725913850668st_nat @ ( produc6126748735280710287st_nat @ Xs @ Ys ) @ ( listrel_c_nat @ R2 ) )
     => ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1068_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_int,R2: set_Pr4413417460947882513_c_int] :
      ( ( member3694854580982123912st_int @ ( produc1948897715771513579st_int @ Xs @ Ys ) @ ( listrel_c_int @ R2 ) )
     => ( ( size_size_list_c @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1069_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_c,R2: set_Product_prod_d_c] :
      ( ( member1739767152416986127list_c @ ( produc242965416994095062list_c @ Xs @ Ys ) @ ( listrel_d_c @ R2 ) )
     => ( ( size_size_list_d @ Xs )
        = ( size_size_list_c @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1070_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( listrel_d_d @ R2 ) )
     => ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1071_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_nat,R2: set_Pr3276347184212964150_d_nat] :
      ( ( member6851837201453759021st_nat @ ( produc6793648210820618640st_nat @ Xs @ Ys ) @ ( listrel_d_nat @ R2 ) )
     => ( ( size_size_list_d @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1072_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_int,R2: set_Pr786264039281237394_d_int] :
      ( ( member4361754056522032265st_int @ ( produc2615797191311421932st_int @ Xs @ Ys ) @ ( listrel_d_int @ R2 ) )
     => ( ( size_size_list_d @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1073_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_c,R2: set_Pr4335409928928841947_nat_c] :
      ( ( member6074218473364765234list_c @ ( produc7723716018658481613list_c @ Xs @ Ys ) @ ( listrel_nat_c @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_c @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1074_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_d,R2: set_Pr4406443968975187932_nat_d] :
      ( ( member6145252513411111219list_d @ ( produc7723716022961710414list_d @ Xs @ Ys ) @ ( listrel_nat_d @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_d @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_1075_distinct__remove1__removeAll,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( remove1_int @ X @ Xs )
        = ( removeAll_int @ X @ Xs ) ) ) ).

% distinct_remove1_removeAll
thf(fact_1076_distinct__remove1__removeAll,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( remove1_nat @ X @ Xs )
        = ( removeAll_nat @ X @ Xs ) ) ) ).

% distinct_remove1_removeAll
thf(fact_1077_mult_Ocomm__neutral,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1078_mult_Ocomm__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1079_mult_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1080_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1081_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_1082_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1083_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1084_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1085_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1086_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1087_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_1088_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% norm_assertion_simps(1)
thf(fact_1089_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% norm_assertion_simps(2)
thf(fact_1090_remove1__idem,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( remove8895909992688781315at_nat @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1091_remove1__idem,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( remove5766854097993331357at_nat @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1092_remove1__idem,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( remove4235003712227595625_nat_o @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1093_remove1__idem,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( remove1_int @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1094_remove1__idem,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( remove1_nat @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1095_remove1__idem,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( remove1670527618125605709n_assn @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1096_notin__set__remove1,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( remove8895909992688781315at_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1097_notin__set__remove1,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( remove5766854097993331357at_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1098_notin__set__remove1,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,Y: produc3658429121746597890et_nat > $o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( remove4235003712227595625_nat_o @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1099_notin__set__remove1,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ~ ( member_int @ X @ ( set_int2 @ ( remove1_int @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1100_notin__set__remove1,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ~ ( member_nat @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1101_notin__set__remove1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( remove1670527618125605709n_assn @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1102_distinct__remove1,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ ( remove1_int @ X @ Xs ) ) ) ).

% distinct_remove1
thf(fact_1103_distinct__remove1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( remove1_nat @ X @ Xs ) ) ) ).

% distinct_remove1
thf(fact_1104_distinct__removeAll,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ ( removeAll_int @ X @ Xs ) ) ) ).

% distinct_removeAll
thf(fact_1105_distinct__removeAll,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).

% distinct_removeAll
thf(fact_1106_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1107_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1108_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1109_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1110_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ one_one_assn )
     => ( entails @ P @ Q ) ) ).

% entails_solve_init(2)
thf(fact_1111_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C4: heap_e7401611519738050253t_unit,D3: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ ( produc7507926704131184380et_nat @ C4 @ D3 ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_1112_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C4: code_integer] : ( P @ ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C4 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_1113_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: heap_e7401611519738050253t_unit,C4: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B4 @ C4 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_1114_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C4: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ C4 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_1115_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C4: int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C4 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_1116_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: int,C4: int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C4 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_1117_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C4: heap_e7401611519738050253t_unit,D3: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ ( produc7507926704131184380et_nat @ C4 @ D3 ) ) ) ) ).

% prod_cases4
thf(fact_1118_prod__cases3,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C4: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C4 ) ) ) ).

% prod_cases3
thf(fact_1119_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: heap_e7401611519738050253t_unit,C4: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B4 @ C4 ) ) ) ).

% prod_cases3
thf(fact_1120_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C4: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ C4 ) ) ) ).

% prod_cases3
thf(fact_1121_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C4: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C4 ) ) ) ).

% prod_cases3
thf(fact_1122_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A4: int > option6357759511663192854e_term,B4: int,C4: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C4 ) ) ) ).

% prod_cases3
thf(fact_1123_Pair__inject,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A7: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A2 @ B2 )
        = ( produc8603105652947943368nteger @ A7 @ B6 ) )
     => ~ ( ( A2 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_1124_Pair__inject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,A7: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A2 @ B2 )
        = ( produc5001842942810119800et_nat @ A7 @ B6 ) )
     => ~ ( ( A2 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_1125_Pair__inject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,A7: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A2 @ B2 )
        = ( produc2245416461498447860et_nat @ A7 @ B6 ) )
     => ~ ( ( A2 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_1126_Pair__inject,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A7: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A2 @ B2 )
        = ( produc5700946648718959541nt_int @ A7 @ B6 ) )
     => ~ ( ( A2 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_1127_Pair__inject,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,A7: int > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A2 @ B2 )
        = ( produc4305682042979456191nt_int @ A7 @ B6 ) )
     => ~ ( ( A2 = A7 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_1128_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P6: produc1908205239877642774nteger] :
      ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A4 @ B4 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_1129_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P6: produc3925858234332021118et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ B4 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_1130_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P6: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ B4 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_1131_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P6: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ B4 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_1132_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P6: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ B4 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_1133_surj__pair,axiom,
    ! [P6: produc1908205239877642774nteger] :
    ? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( P6
      = ( produc8603105652947943368nteger @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_1134_surj__pair,axiom,
    ! [P6: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
      ( P6
      = ( produc5001842942810119800et_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_1135_surj__pair,axiom,
    ! [P6: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
      ( P6
      = ( produc2245416461498447860et_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_1136_surj__pair,axiom,
    ! [P6: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( P6
      = ( produc5700946648718959541nt_int @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_1137_surj__pair,axiom,
    ! [P6: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( P6
      = ( produc4305682042979456191nt_int @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_1138_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1139_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1140_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1141_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1142_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1143_remove__code_I1_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( remove_int @ X @ ( set_int2 @ Xs ) )
      = ( set_int2 @ ( removeAll_int @ X @ Xs ) ) ) ).

% remove_code(1)
thf(fact_1144_remove__code_I1_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( remove_nat @ X @ ( set_nat2 @ Xs ) )
      = ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) ) ).

% remove_code(1)
thf(fact_1145_remove__code_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( remove6142905432236673558n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
      = ( set_Pr1139785259514867910n_assn @ ( remove1763081955827946941n_assn @ X @ Xs ) ) ) ).

% remove_code(1)
thf(fact_1146_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_1147_length__removeAll__less,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ord_less_nat @ ( size_s1588839187904960729at_nat @ ( remove4908499291543849587at_nat @ X @ Xs ) ) @ ( size_s1588839187904960729at_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1148_length__removeAll__less,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ ( size_s5460976970255530739at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) ) @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1149_length__removeAll__less,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ord_less_nat @ ( size_s8908918004648414271_nat_o @ ( remove7604696131425036249_nat_o @ X @ Xs ) ) @ ( size_s8908918004648414271_nat_o @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1150_length__removeAll__less,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ord_less_nat @ ( size_s6829681357464350627n_assn @ ( remove1763081955827946941n_assn @ X @ Xs ) ) @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1151_length__removeAll__less,axiom,
    ! [X: c,Xs: list_c] :
      ( ( member_c @ X @ ( set_c2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_c @ ( removeAll_c @ X @ Xs ) ) @ ( size_size_list_c @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1152_length__removeAll__less,axiom,
    ! [X: d,Xs: list_d] :
      ( ( member_d @ X @ ( set_d2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_d @ ( removeAll_d @ X @ Xs ) ) @ ( size_size_list_d @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1153_length__removeAll__less,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1154_length__removeAll__less,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_1155_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1156_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_1157_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1158_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1159_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( ( As = bot_bot_set_nat )
              & X ) ) ) ).

% pure_assn_raw.elims(3)
thf(fact_1160_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ~ ( ( As = bot_bot_set_nat )
                & X ) ) ) ).

% pure_assn_raw.elims(2)
thf(fact_1161_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( ~ ( ( As = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.elims(1)
thf(fact_1162_pure__assn__raw_Osimps,axiom,
    ! [B2: $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( pure_a825153325127701367it_nat @ B2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ( As2 = bot_bot_set_nat )
        & B2 ) ) ).

% pure_assn_raw.simps
thf(fact_1163_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_c] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Ys ) )
       => ( ( nth_Product_prod_c_c @ ( zip_c_c @ Xs @ Ys ) @ I )
          = ( product_Pair_c_c @ ( nth_c @ Xs @ I ) @ ( nth_c @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1164_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_d] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Ys ) )
       => ( ( nth_Product_prod_c_d @ ( zip_c_d @ Xs @ Ys ) @ I )
          = ( product_Pair_c_d @ ( nth_c @ Xs @ I ) @ ( nth_d @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1165_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1708982275578427192_c_nat @ ( zip_c_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_c_nat @ ( nth_c @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1166_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_c @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr6754503292924006292_c_int @ ( zip_c_int @ Xs @ Ys ) @ I )
          = ( product_Pair_c_int @ ( nth_c @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1167_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_c] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Ys ) )
       => ( ( nth_Product_prod_d_c @ ( zip_d_c @ Xs @ Ys ) @ I )
          = ( product_Pair_d_c @ ( nth_d @ Xs @ I ) @ ( nth_c @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1168_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_d] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Ys ) )
       => ( ( nth_Product_prod_d_d @ ( zip_d_d @ Xs @ Ys ) @ I )
          = ( product_Pair_d_d @ ( nth_d @ Xs @ I ) @ ( nth_d @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1169_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr2944426604534821689_d_nat @ ( zip_d_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_d_nat @ ( nth_d @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1170_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_d @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr7989947621880400789_d_int @ ( zip_d_int @ Xs @ Ys ) @ I )
          = ( product_Pair_d_int @ ( nth_d @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1171_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_c] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_c @ Ys ) )
       => ( ( nth_Pr2476257089995772766_nat_c @ ( zip_nat_c @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_c @ ( nth_nat @ Xs @ I ) @ ( nth_c @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1172_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_d] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_d @ Ys ) )
       => ( ( nth_Pr2476257094299001567_nat_d @ ( zip_nat_d @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_d @ ( nth_nat @ Xs @ I ) @ ( nth_d @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1173_curryI,axiom,
    ! [F4: produc1908205239877642774nteger > $o,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( F4 @ ( produc8603105652947943368nteger @ A2 @ B2 ) )
     => ( produc1217013493180205570eger_o @ F4 @ A2 @ B2 ) ) ).

% curryI
thf(fact_1174_curryI,axiom,
    ! [F4: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( F4 @ ( produc5001842942810119800et_nat @ A2 @ B2 ) )
     => ( produc6216949301066131538_nat_o @ F4 @ A2 @ B2 ) ) ).

% curryI
thf(fact_1175_curryI,axiom,
    ! [F4: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( F4 @ ( produc2245416461498447860et_nat @ A2 @ B2 ) )
     => ( produc5101573711933517782_nat_o @ F4 @ A2 @ B2 ) ) ).

% curryI
thf(fact_1176_curryI,axiom,
    ! [F4: produc2285326912895808259nt_int > $o,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( F4 @ ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ( produc730925184835016917_int_o @ F4 @ A2 @ B2 ) ) ).

% curryI
thf(fact_1177_curryI,axiom,
    ! [F4: produc7773217078559923341nt_int > $o,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( F4 @ ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ( produc9098658269643458507_int_o @ F4 @ A2 @ B2 ) ) ).

% curryI
thf(fact_1178_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_c,A7: list_c,B6: list_c] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_c_c @ A2 @ B2 )
            = ( zip_c_c @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1179_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_d,A7: list_c,B6: list_d] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_c_d @ A2 @ B2 )
            = ( zip_c_d @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1180_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_nat,A7: list_c,B6: list_nat] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_nat @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_c_nat @ A2 @ B2 )
            = ( zip_c_nat @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1181_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_int,A7: list_c,B6: list_int] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_int @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_c_int @ A2 @ B2 )
            = ( zip_c_int @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1182_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_c,A7: list_d,B6: list_c] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_d_c @ A2 @ B2 )
            = ( zip_d_c @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1183_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_d,A7: list_d,B6: list_d] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_d_d @ A2 @ B2 )
            = ( zip_d_d @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1184_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_nat,A7: list_d,B6: list_nat] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_nat @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_d_nat @ A2 @ B2 )
            = ( zip_d_nat @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1185_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_int,A7: list_d,B6: list_int] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_int @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_d_int @ A2 @ B2 )
            = ( zip_d_int @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1186_zip__eq__zip__same__len,axiom,
    ! [A2: list_nat,B2: list_c,A7: list_nat,B6: list_c] :
      ( ( ( size_size_list_nat @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_nat @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_nat_c @ A2 @ B2 )
            = ( zip_nat_c @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1187_zip__eq__zip__same__len,axiom,
    ! [A2: list_nat,B2: list_d,A7: list_nat,B6: list_d] :
      ( ( ( size_size_list_nat @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_nat @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_nat_d @ A2 @ B2 )
            = ( zip_nat_d @ A7 @ B6 ) )
          = ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_1188_zip__inj,axiom,
    ! [A2: list_c,B2: list_c,A7: list_c,B6: list_c] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_c_c @ A2 @ B2 )
            = ( zip_c_c @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1189_zip__inj,axiom,
    ! [A2: list_c,B2: list_d,A7: list_c,B6: list_d] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_c_d @ A2 @ B2 )
            = ( zip_c_d @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1190_zip__inj,axiom,
    ! [A2: list_c,B2: list_nat,A7: list_c,B6: list_nat] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_nat @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_c_nat @ A2 @ B2 )
            = ( zip_c_nat @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1191_zip__inj,axiom,
    ! [A2: list_c,B2: list_int,A7: list_c,B6: list_int] :
      ( ( ( size_size_list_c @ A2 )
        = ( size_size_list_int @ B2 ) )
     => ( ( ( size_size_list_c @ A7 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_c_int @ A2 @ B2 )
            = ( zip_c_int @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1192_zip__inj,axiom,
    ! [A2: list_d,B2: list_c,A7: list_d,B6: list_c] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_d_c @ A2 @ B2 )
            = ( zip_d_c @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1193_zip__inj,axiom,
    ! [A2: list_d,B2: list_d,A7: list_d,B6: list_d] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_d_d @ A2 @ B2 )
            = ( zip_d_d @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1194_zip__inj,axiom,
    ! [A2: list_d,B2: list_nat,A7: list_d,B6: list_nat] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_nat @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_d_nat @ A2 @ B2 )
            = ( zip_d_nat @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1195_zip__inj,axiom,
    ! [A2: list_d,B2: list_int,A7: list_d,B6: list_int] :
      ( ( ( size_size_list_d @ A2 )
        = ( size_size_list_int @ B2 ) )
     => ( ( ( size_size_list_d @ A7 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_d_int @ A2 @ B2 )
            = ( zip_d_int @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1196_zip__inj,axiom,
    ! [A2: list_nat,B2: list_c,A7: list_nat,B6: list_c] :
      ( ( ( size_size_list_nat @ A2 )
        = ( size_size_list_c @ B2 ) )
     => ( ( ( size_size_list_nat @ A7 )
          = ( size_size_list_c @ B6 ) )
       => ( ( ( zip_nat_c @ A2 @ B2 )
            = ( zip_nat_c @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1197_zip__inj,axiom,
    ! [A2: list_nat,B2: list_d,A7: list_nat,B6: list_d] :
      ( ( ( size_size_list_nat @ A2 )
        = ( size_size_list_d @ B2 ) )
     => ( ( ( size_size_list_nat @ A7 )
          = ( size_size_list_d @ B6 ) )
       => ( ( ( zip_nat_d @ A2 @ B2 )
            = ( zip_nat_d @ A7 @ B6 ) )
         => ( ( A2 = A7 )
            & ( B2 = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_1198_pair__list__split,axiom,
    ! [L: list_P267710562049004973od_c_c] :
      ~ ! [L1: list_c,L22: list_c] :
          ( ( L
            = ( zip_c_c @ L1 @ L22 ) )
         => ( ( ( size_size_list_c @ L1 )
              = ( size_size_list_c @ L22 ) )
           => ( ( size_s2756448709718749209od_c_c @ L )
             != ( size_size_list_c @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1199_pair__list__split,axiom,
    ! [L: list_P338744602095350958od_c_d] :
      ~ ! [L1: list_c,L22: list_d] :
          ( ( L
            = ( zip_c_d @ L1 @ L22 ) )
         => ( ( ( size_size_list_c @ L1 )
              = ( size_size_list_d @ L22 ) )
           => ( ( size_s2827482749765095194od_c_d @ L )
             != ( size_size_list_d @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1200_pair__list__split,axiom,
    ! [L: list_P5561950507774946575_c_nat] :
      ~ ! [L1: list_c,L22: list_nat] :
          ( ( L
            = ( zip_c_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_c @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s2954062820725854115_c_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1201_pair__list__split,axiom,
    ! [L: list_P3071867362843219819_c_int] :
      ~ ! [L1: list_c,L22: list_int] :
          ( ( L
            = ( zip_c_int @ L1 @ L22 ) )
         => ( ( ( size_size_list_c @ L1 )
              = ( size_size_list_int @ L22 ) )
           => ( ( size_s463979675794127359_c_int @ L )
             != ( size_size_list_int @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1202_pair__list__split,axiom,
    ! [L: list_P8855433598298794092od_d_c] :
      ~ ! [L1: list_d,L22: list_c] :
          ( ( L
            = ( zip_d_c @ L1 @ L22 ) )
         => ( ( ( size_size_list_d @ L1 )
              = ( size_size_list_c @ L22 ) )
           => ( ( size_s2120799709113762520od_d_c @ L )
             != ( size_size_list_c @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1203_pair__list__split,axiom,
    ! [L: list_P8926467638345140077od_d_d] :
      ~ ! [L1: list_d,L22: list_d] :
          ( ( L
            = ( zip_d_d @ L1 @ L22 ) )
         => ( ( ( size_size_list_d @ L1 )
              = ( size_size_list_d @ L22 ) )
           => ( ( size_s2191833749160108505od_d_d @ L )
             != ( size_size_list_d @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1204_pair__list__split,axiom,
    ! [L: list_P1934797086108301456_d_nat] :
      ~ ! [L1: list_d,L22: list_nat] :
          ( ( L
            = ( zip_d_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_d @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s8550281435913984804_d_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1205_pair__list__split,axiom,
    ! [L: list_P8668085978031350508_d_int] :
      ~ ! [L1: list_d,L22: list_int] :
          ( ( L
            = ( zip_d_int @ L1 @ L22 ) )
         => ( ( ( size_size_list_d @ L1 )
              = ( size_size_list_int @ L22 ) )
           => ( ( size_s6060198290982258048_d_int @ L )
             != ( size_size_list_int @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1206_pair__list__split,axiom,
    ! [L: list_P2993859830824179253_nat_c] :
      ~ ! [L1: list_nat,L22: list_c] :
          ( ( L
            = ( zip_nat_c @ L1 @ L22 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_c @ L22 ) )
           => ( ( size_s385972143775086793_nat_c @ L )
             != ( size_size_list_c @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1207_pair__list__split,axiom,
    ! [L: list_P3064893870870525238_nat_d] :
      ~ ! [L1: list_nat,L22: list_d] :
          ( ( L
            = ( zip_nat_d @ L1 @ L22 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_d @ L22 ) )
           => ( ( size_s457006183821432778_nat_d @ L )
             != ( size_size_list_d @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_1208_curryD,axiom,
    ! [F4: produc1908205239877642774nteger > $o,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc8603105652947943368nteger @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_1209_curryD,axiom,
    ! [F4: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc5001842942810119800et_nat @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_1210_curryD,axiom,
    ! [F4: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc2245416461498447860et_nat @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_1211_curryD,axiom,
    ! [F4: produc2285326912895808259nt_int > $o,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc5700946648718959541nt_int @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_1212_curryD,axiom,
    ! [F4: produc7773217078559923341nt_int > $o,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc4305682042979456191nt_int @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_1213_curryE,axiom,
    ! [F4: produc1908205239877642774nteger > $o,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc8603105652947943368nteger @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_1214_curryE,axiom,
    ! [F4: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc5001842942810119800et_nat @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_1215_curryE,axiom,
    ! [F4: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc2245416461498447860et_nat @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_1216_curryE,axiom,
    ! [F4: produc2285326912895808259nt_int > $o,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc5700946648718959541nt_int @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_1217_curryE,axiom,
    ! [F4: produc7773217078559923341nt_int > $o,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F4 @ A2 @ B2 )
     => ( F4 @ ( produc4305682042979456191nt_int @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_1218_set__zip__rightD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Ys ) ) )
     => ( member2643936169264416010at_nat @ Y @ ( set_se5049602875457034614at_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1219_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1220_set__zip__rightD,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ X @ Y ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Ys ) ) )
     => ( member_assn @ Y @ ( set_assn2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1221_set__zip__rightD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
     => ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1222_set__zip__rightD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( set_Pr364071364088684201et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) ) )
     => ( member6260224972018164377et_nat @ Y @ ( set_Pr3864771053587467565et_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1223_set__zip__rightD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( set_Pr7458301993993767461et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) ) )
     => ( member1996754912294343701et_nat @ Y @ ( set_Pr364071364088684201et_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1224_set__zip__rightD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1225_set__zip__rightD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1226_set__zip__leftD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Ys ) ) )
     => ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1227_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1228_set__zip__leftD,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ X @ Y ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Ys ) ) )
     => ( member_assn @ X @ ( set_assn2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1229_set__zip__leftD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
     => ( member4242434998011752849e_term @ X @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1230_set__zip__leftD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( set_Pr364071364088684201et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1231_set__zip__leftD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( set_Pr7458301993993767461et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1232_set__zip__leftD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1233_set__zip__leftD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1234_in__set__zipE,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1235_in__set__zipE,axiom,
    ! [X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1236_in__set__zipE,axiom,
    ! [X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( set_Pr1470767568048878706at_int @ ( zip_nat_int @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1237_in__set__zipE,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1238_in__set__zipE,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ X @ Y ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Ys ) ) )
     => ~ ( ( member_assn @ X @ ( set_assn2 @ Xs ) )
         => ~ ( member_assn @ Y @ ( set_assn2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1239_in__set__zipE,axiom,
    ! [X: product_prod_nat_nat,Y: int,Xs: list_P6011104703257516679at_nat,Ys: list_int] :
      ( ( member8394280151737582451at_int @ ( produc6348220600061155286at_int @ X @ Y ) @ ( set_Pr3658594827066574855at_int @ ( zip_Pr6866960147343648950at_int @ Xs @ Ys ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1240_in__set__zipE,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X @ Y ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1241_in__set__zipE,axiom,
    ! [X: int,Y: product_prod_nat_nat,Xs: list_int,Ys: list_P6011104703257516679at_nat] :
      ( ( member5261114234888836065at_nat @ ( produc3525228511222278652at_nat @ X @ Y ) @ ( set_Pr525428910217828469at_nat @ ( zip_in4043968058504772316at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1242_in__set__zipE,axiom,
    ! [X: int,Y: produc6575502325842934193n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( member241480614033477777n_assn @ ( produc3313821056773121708n_assn @ X @ Y ) @ ( set_Pr4510407026980144421n_assn @ ( zip_in7098856637274830732n_assn @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
         => ~ ( member7957490590177025114n_assn @ Y @ ( set_Pr1139785259514867910n_assn @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1243_in__set__zipE,axiom,
    ! [X: nat,Y: product_prod_nat_nat,Xs: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X @ Y ) @ ( set_Pr6710958862608470481at_nat @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1244_zip__same,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A2 @ B2 ) @ ( set_Pr4552580085251402108at_nat @ ( zip_Pr2522280425463452279at_nat @ Xs @ Xs ) ) )
      = ( ( member8757157785044589968at_nat @ A2 @ ( set_Pr3765526544606949372at_nat @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1245_zip__same,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1246_zip__same,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ ( set_Pr8666615595052072444_nat_o @ ( zip_Pr2849639184604596727_nat_o @ Xs @ Xs ) ) )
      = ( ( member6576561426505652726_nat_o @ A2 @ ( set_Pr400311997697478754_nat_o @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1247_zip__same,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Xs ) ) )
      = ( ( member2643936169264416010at_nat @ A2 @ ( set_se5049602875457034614at_nat @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1248_zip__same,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
      = ( ( member_int @ A2 @ ( set_int2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1249_zip__same,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1250_zip__same,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ A2 @ B2 ) @ ( set_Pr8975451644279219196n_assn @ ( zip_Pr8806919808394484215n_assn @ Xs @ Xs ) ) )
      = ( ( member7957490590177025114n_assn @ A2 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1251_zip__same,axiom,
    ! [A2: assn,B2: assn,Xs: list_assn] :
      ( ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ A2 @ B2 ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Xs ) ) )
      = ( ( member_assn @ A2 @ ( set_assn2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1252_zip__update,axiom,
    ! [Xs: list_c,I: nat,X: c,Ys: list_c,Y: c] :
      ( ( zip_c_c @ ( list_update_c @ Xs @ I @ X ) @ ( list_update_c @ Ys @ I @ Y ) )
      = ( list_u8616212504517173525od_c_c @ ( zip_c_c @ Xs @ Ys ) @ I @ ( product_Pair_c_c @ X @ Y ) ) ) ).

% zip_update
thf(fact_1253_zip__update,axiom,
    ! [Xs: list_c,I: nat,X: c,Ys: list_d,Y: d] :
      ( ( zip_c_d @ ( list_update_c @ Xs @ I @ X ) @ ( list_update_d @ Ys @ I @ Y ) )
      = ( list_u8616212508820402326od_c_d @ ( zip_c_d @ Xs @ Ys ) @ I @ ( product_Pair_c_d @ X @ Y ) ) ) ).

% zip_update
thf(fact_1254_zip__update,axiom,
    ! [Xs: list_d,I: nat,X: d,Ys: list_c,Y: c] :
      ( ( zip_d_c @ ( list_update_d @ Xs @ I @ X ) @ ( list_update_c @ Ys @ I @ Y ) )
      = ( list_u5828756922544497492od_d_c @ ( zip_d_c @ Xs @ Ys ) @ I @ ( product_Pair_d_c @ X @ Y ) ) ) ).

% zip_update
thf(fact_1255_zip__update,axiom,
    ! [Xs: list_d,I: nat,X: d,Ys: list_d,Y: d] :
      ( ( zip_d_d @ ( list_update_d @ Xs @ I @ X ) @ ( list_update_d @ Ys @ I @ Y ) )
      = ( list_u5828756926847726293od_d_d @ ( zip_d_d @ Xs @ Ys ) @ I @ ( product_Pair_d_d @ X @ Y ) ) ) ).

% zip_update
thf(fact_1256_zip__update,axiom,
    ! [Xs: list_P1316552470764441098e_term,I: nat,X: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
      ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y ) )
      = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ).

% zip_update
thf(fact_1257_zip__update,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,I: nat,X: produc3658429121746597890et_nat > $o,Ys: list_P9062070895058802706et_nat,Y: produc3658429121746597890et_nat] :
      ( ( zip_Pr7134870689397686104et_nat @ ( list_u6943956310655620667_nat_o @ Xs @ I @ X ) @ ( list_u9060326803697358356et_nat @ Ys @ I @ Y ) )
      = ( list_u1866171116859985808et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) @ I @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1258_zip__update,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,I: nat,X: produc3658429121746597890et_nat > $o,Ys: list_P2321686559999237006et_nat,Y: produc3925858234332021118et_nat] :
      ( ( zip_Pr8136144321567152340et_nat @ ( list_u6943956310655620667_nat_o @ Xs @ I @ X ) @ ( list_u1866171116859985808et_nat @ Ys @ I @ Y ) )
      = ( list_u1913845194572276492et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) @ I @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1259_zip__update,axiom,
    ! [Xs: list_P1743416141875011707e_term,I: nat,X: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
      = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1260_zip__update,axiom,
    ! [Xs: list_i8448526496819171953e_term,I: nat,X: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y ) )
      = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1261_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_assn,Ys: list_assn,X: assn] :
      ( ( ( size_size_list_assn @ Xs )
        = ( size_size_list_assn @ Ys ) )
     => ( ( member_assn @ X @ ( set_assn2 @ Xs ) )
       => ~ ! [Y2: assn] :
              ~ ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ X @ Y2 ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1262_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_c,X: c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ~ ! [Y2: c] :
              ~ ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X @ Y2 ) @ ( set_Product_prod_c_c2 @ ( zip_c_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1263_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_d,X: c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ~ ! [Y2: d] :
              ~ ( member5074992363344545361od_c_d @ ( product_Pair_c_d @ X @ Y2 ) @ ( set_Product_prod_c_d2 @ ( zip_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1264_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_nat,X: c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ~ ! [Y2: nat] :
              ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y2 ) @ ( set_Pr3395872032415823530_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1265_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_int,X: c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ~ ! [Y2: int] :
              ~ ( member4017226226790010994_c_int @ ( product_Pair_c_int @ X @ Y2 ) @ ( set_Pr8441393049761402630_c_int @ ( zip_c_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1266_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_c,X: d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ~ ! [Y2: c] :
              ~ ( member2287536777068640527od_d_c @ ( product_Pair_d_c @ X @ Y2 ) @ ( set_Product_prod_d_c2 @ ( zip_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1267_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_d,X: d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ~ ! [Y2: d] :
              ~ ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X @ Y2 ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1268_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_nat,X: d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ~ ! [Y2: nat] :
              ~ ( member207149538400826391_d_nat @ ( product_Pair_d_nat @ X @ Y2 ) @ ( set_Pr4631316361372218027_d_nat @ ( zip_d_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1269_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_int,X: d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ~ ! [Y2: int] :
              ~ ( member5252670555746405491_d_int @ ( product_Pair_d_int @ X @ Y2 ) @ ( set_Pr453465341863021319_d_int @ ( zip_d_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1270_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_c,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y2: c] :
              ~ ( member8962352060716553276_nat_c @ ( product_Pair_nat_c @ X @ Y2 ) @ ( set_Pr4163146846833169104_nat_c @ ( zip_nat_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1271_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_assn,Ys: list_assn,Y: assn] :
      ( ( ( size_size_list_assn @ Xs )
        = ( size_size_list_assn @ Ys ) )
     => ( ( member_assn @ Y @ ( set_assn2 @ Ys ) )
       => ~ ! [X3: assn] :
              ~ ( member7957490590177025114n_assn @ ( produc118845697133431529n_assn @ X3 @ Y ) @ ( set_Pr1139785259514867910n_assn @ ( zip_assn_assn @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1272_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_c,Y: c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_c @ Y @ ( set_c2 @ Ys ) )
       => ~ ! [X3: c] :
              ~ ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X3 @ Y ) @ ( set_Product_prod_c_c2 @ ( zip_c_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1273_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_d,Y: d] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( member_d @ Y @ ( set_d2 @ Ys ) )
       => ~ ! [X3: c] :
              ~ ( member5074992363344545361od_c_d @ ( product_Pair_c_d @ X3 @ Y ) @ ( set_Product_prod_c_d2 @ ( zip_c_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1274_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: c] :
              ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y ) @ ( set_Pr3395872032415823530_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1275_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_int,Y: int] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: c] :
              ~ ( member4017226226790010994_c_int @ ( product_Pair_c_int @ X3 @ Y ) @ ( set_Pr8441393049761402630_c_int @ ( zip_c_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1276_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_c,Y: c] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_c @ Y @ ( set_c2 @ Ys ) )
       => ~ ! [X3: d] :
              ~ ( member2287536777068640527od_d_c @ ( product_Pair_d_c @ X3 @ Y ) @ ( set_Product_prod_d_c2 @ ( zip_d_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1277_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_d,Y: d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( member_d @ Y @ ( set_d2 @ Ys ) )
       => ~ ! [X3: d] :
              ~ ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X3 @ Y ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1278_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: d] :
              ~ ( member207149538400826391_d_nat @ ( product_Pair_d_nat @ X3 @ Y ) @ ( set_Pr4631316361372218027_d_nat @ ( zip_d_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1279_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_int,Y: int] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: d] :
              ~ ( member5252670555746405491_d_int @ ( product_Pair_d_int @ X3 @ Y ) @ ( set_Pr453465341863021319_d_int @ ( zip_d_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1280_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_c,Y: c] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( member_c @ Y @ ( set_c2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member8962352060716553276_nat_c @ ( product_Pair_nat_c @ X3 @ Y ) @ ( set_Pr4163146846833169104_nat_c @ ( zip_nat_c @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1281_length__remove1,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
       => ( ( size_s1588839187904960729at_nat @ ( remove8895909992688781315at_nat @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s1588839187904960729at_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
       => ( ( size_s1588839187904960729at_nat @ ( remove8895909992688781315at_nat @ X @ Xs ) )
          = ( size_s1588839187904960729at_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1282_length__remove1,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( ( size_s5460976970255530739at_nat @ ( remove5766854097993331357at_nat @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( ( size_s5460976970255530739at_nat @ ( remove5766854097993331357at_nat @ X @ Xs ) )
          = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1283_length__remove1,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
       => ( ( size_s8908918004648414271_nat_o @ ( remove4235003712227595625_nat_o @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s8908918004648414271_nat_o @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
       => ( ( size_s8908918004648414271_nat_o @ ( remove4235003712227595625_nat_o @ X @ Xs ) )
          = ( size_s8908918004648414271_nat_o @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1284_length__remove1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
       => ( ( size_s6829681357464350627n_assn @ ( remove1670527618125605709n_assn @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s6829681357464350627n_assn @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
       => ( ( size_s6829681357464350627n_assn @ ( remove1670527618125605709n_assn @ X @ Xs ) )
          = ( size_s6829681357464350627n_assn @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1285_length__remove1,axiom,
    ! [X: c,Xs: list_c] :
      ( ( ( member_c @ X @ ( set_c2 @ Xs ) )
       => ( ( size_size_list_c @ ( remove1_c @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_c @ X @ ( set_c2 @ Xs ) )
       => ( ( size_size_list_c @ ( remove1_c @ X @ Xs ) )
          = ( size_size_list_c @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1286_length__remove1,axiom,
    ! [X: d,Xs: list_d] :
      ( ( ( member_d @ X @ ( set_d2 @ Xs ) )
       => ( ( size_size_list_d @ ( remove1_d @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_d @ X @ ( set_d2 @ Xs ) )
       => ( ( size_size_list_d @ ( remove1_d @ X @ Xs ) )
          = ( size_size_list_d @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1287_length__remove1,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1288_length__remove1,axiom,
    ! [X: int,Xs: list_int] :
      ( ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( ( size_size_list_int @ ( remove1_int @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( ( size_size_list_int @ ( remove1_int @ X @ Xs ) )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1289_insert__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( insert_nat2 @ X @ ( coset_nat @ Xs ) )
      = ( coset_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).

% insert_code(2)
thf(fact_1290_insert__code_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( insert_int2 @ X @ ( coset_int @ Xs ) )
      = ( coset_int @ ( removeAll_int @ X @ Xs ) ) ) ).

% insert_code(2)
thf(fact_1291_insert__code_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( coset_6809516720700967924at_nat @ Xs ) )
      = ( coset_6809516720700967924at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) ) ) ).

% insert_code(2)
thf(fact_1292_insert__code_I2_J,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( coset_5307619257043544794at_nat @ Xs ) )
      = ( coset_5307619257043544794at_nat @ ( remove4908499291543849587at_nat @ X @ Xs ) ) ) ).

% insert_code(2)
thf(fact_1293_insert__code_I2_J,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( insert_o2 @ X @ ( coset_o @ Xs ) )
      = ( coset_o @ ( removeAll_o @ X @ Xs ) ) ) ).

% insert_code(2)
thf(fact_1294_zipf__zip,axiom,
    ! [L12: list_c,L23: list_c] :
      ( ( ( size_size_list_c @ L12 )
        = ( size_size_list_c @ L23 ) )
     => ( ( zipf_c684006316856269033od_c_c @ product_Pair_c_c @ L12 @ L23 )
        = ( zip_c_c @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1295_zipf__zip,axiom,
    ! [L12: list_c,L23: list_d] :
      ( ( ( size_size_list_c @ L12 )
        = ( size_size_list_d @ L23 ) )
     => ( ( zipf_c44566020827681705od_c_d @ product_Pair_c_d @ L12 @ L23 )
        = ( zip_c_d @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1296_zipf__zip,axiom,
    ! [L12: list_c,L23: list_nat] :
      ( ( ( size_size_list_c @ L12 )
        = ( size_size_list_nat @ L23 ) )
     => ( ( zipf_c7084520048314076179_c_nat @ product_Pair_c_nat @ L12 @ L23 )
        = ( zip_c_nat @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1297_zipf__zip,axiom,
    ! [L12: list_c,L23: list_int] :
      ( ( ( size_size_list_c @ L12 )
        = ( size_size_list_int @ L23 ) )
     => ( ( zipf_c230841880303976139_c_int @ product_Pair_c_int @ L12 @ L23 )
        = ( zip_c_int @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1298_zipf__zip,axiom,
    ! [L12: list_d,L23: list_c] :
      ( ( ( size_size_list_d @ L12 )
        = ( size_size_list_c @ L23 ) )
     => ( ( zipf_d1836499321508429225od_d_c @ product_Pair_d_c @ L12 @ L23 )
        = ( zip_d_c @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1299_zipf__zip,axiom,
    ! [L12: list_d,L23: list_d] :
      ( ( ( size_size_list_d @ L12 )
        = ( size_size_list_d @ L23 ) )
     => ( ( zipf_d1197059025479841897od_d_d @ product_Pair_d_d @ L12 @ L23 )
        = ( zip_d_d @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1300_zipf__zip,axiom,
    ! [L12: list_d,L23: list_nat] :
      ( ( ( size_size_list_d @ L12 )
        = ( size_size_list_nat @ L23 ) )
     => ( ( zipf_d7851395264309270293_d_nat @ product_Pair_d_nat @ L12 @ L23 )
        = ( zip_d_nat @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1301_zipf__zip,axiom,
    ! [L12: list_d,L23: list_int] :
      ( ( ( size_size_list_d @ L12 )
        = ( size_size_list_int @ L23 ) )
     => ( ( zipf_d997717096299170253_d_int @ product_Pair_d_int @ L12 @ L23 )
        = ( zip_d_int @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1302_zipf__zip,axiom,
    ! [L12: list_nat,L23: list_c] :
      ( ( ( size_size_list_nat @ L12 )
        = ( size_size_list_c @ L23 ) )
     => ( ( zipf_n5773672615478942615_nat_c @ product_Pair_nat_c @ L12 @ L23 )
        = ( zip_nat_c @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1303_zipf__zip,axiom,
    ! [L12: list_nat,L23: list_d] :
      ( ( ( size_size_list_nat @ L12 )
        = ( size_size_list_d @ L23 ) )
     => ( ( zipf_n3657244561791696409_nat_d @ product_Pair_nat_d @ L12 @ L23 )
        = ( zip_nat_d @ L12 @ L23 ) ) ) ).

% zipf_zip
thf(fact_1304_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_1305_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_1306_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_1307_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_1308_psubset__insert__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2453136405763929_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ B )
         => ( ord_le2453136405763929_nat_o @ A @ B ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ B )
         => ( ( ( member6576561426505652726_nat_o @ X @ A )
             => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B ) )
            & ( ~ ( member6576561426505652726_nat_o @ X @ A )
             => ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1309_psubset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
      = ( ( ( member8757157785044589968at_nat @ X @ B )
         => ( ord_le2604355607129572851at_nat @ A @ B ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ B )
         => ( ( ( member8757157785044589968at_nat @ X @ A )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B ) )
            & ( ~ ( member8757157785044589968at_nat @ X @ A )
             => ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1310_psubset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B )
         => ( ord_le7866589430770878221at_nat @ A @ B ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B )
         => ( ( ( member8440522571783428010at_nat @ X @ A )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A )
             => ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1311_psubset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_set_o @ A @ ( insert_o2 @ X @ B ) )
      = ( ( ( member_o @ X @ B )
         => ( ord_less_set_o @ A @ B ) )
        & ( ~ ( member_o @ X @ B )
         => ( ( ( member_o @ X @ A )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B ) )
            & ( ~ ( member_o @ X @ A )
             => ( ord_less_eq_set_o @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1312_psubset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_set_int @ A @ ( insert_int2 @ X @ B ) )
      = ( ( ( member_int @ X @ B )
         => ( ord_less_set_int @ A @ B ) )
        & ( ~ ( member_int @ X @ B )
         => ( ( ( member_int @ X @ A )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B ) )
            & ( ~ ( member_int @ X @ A )
             => ( ord_less_eq_set_int @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1313_psubset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ ( insert_nat2 @ X @ B ) )
      = ( ( ( member_nat @ X @ B )
         => ( ord_less_set_nat @ A @ B ) )
        & ( ~ ( member_nat @ X @ B )
         => ( ( ( member_nat @ X @ A )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
            & ( ~ ( member_nat @ X @ A )
             => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1314_lexn__length,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c,N: nat] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( lexn_c @ R2 @ N ) )
     => ( ( ( size_size_list_c @ Xs )
          = N )
        & ( ( size_size_list_c @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_1315_lexn__length,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d,N: nat] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( lexn_d @ R2 @ N ) )
     => ( ( ( size_size_list_d @ Xs )
          = N )
        & ( ( size_size_list_d @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_1316_lexn__length,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexn_nat @ R2 @ N ) )
     => ( ( ( size_size_list_nat @ Xs )
          = N )
        & ( ( size_size_list_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_1317_lexn__length,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,N: nat] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lexn_int @ R2 @ N ) )
     => ( ( ( size_size_list_int @ Xs )
          = N )
        & ( ( size_size_list_int @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_1318_pairwise__alt,axiom,
    ( pairwi3222497012029234898at_nat
    = ( ^ [R3: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,S2: set_Pr4329608150637261639at_nat] :
        ! [X2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ S2 )
         => ! [Y5: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Y5 @ ( minus_3314409938677909166at_nat @ S2 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
             => ( R3 @ X2 @ Y5 ) ) ) ) ) ).

% pairwise_alt
thf(fact_1319_pairwise__alt,axiom,
    ( pairwi6357175311820755948at_nat
    = ( ^ [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S2: set_Pr1261947904930325089at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ S2 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
             => ( R3 @ X2 @ Y5 ) ) ) ) ) ).

% pairwise_alt
thf(fact_1320_pairwise__alt,axiom,
    ( pairwise_o
    = ( ^ [R3: $o > $o > $o,S2: set_o] :
        ! [X2: $o] :
          ( ( member_o @ X2 @ S2 )
         => ! [Y5: $o] :
              ( ( member_o @ Y5 @ ( minus_minus_set_o @ S2 @ ( insert_o2 @ X2 @ bot_bot_set_o ) ) )
             => ( R3 @ X2 @ Y5 ) ) ) ) ) ).

% pairwise_alt
thf(fact_1321_pairwise__alt,axiom,
    ( pairwise_int
    = ( ^ [R3: int > int > $o,S2: set_int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ S2 )
         => ! [Y5: int] :
              ( ( member_int @ Y5 @ ( minus_minus_set_int @ S2 @ ( insert_int2 @ X2 @ bot_bot_set_int ) ) )
             => ( R3 @ X2 @ Y5 ) ) ) ) ) ).

% pairwise_alt
thf(fact_1322_pairwise__alt,axiom,
    ( pairwise_nat
    = ( ^ [R3: nat > nat > $o,S2: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ S2 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ ( minus_minus_set_nat @ S2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
             => ( R3 @ X2 @ Y5 ) ) ) ) ) ).

% pairwise_alt
thf(fact_1323_and__extract__pure__left__iff,axiom,
    ! [B2: $o,Q: assn] :
      ( ( inf_inf_assn @ ( pure_assn @ B2 ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ Q ) @ ( pure_assn @ B2 ) ) ) ).

% and_extract_pure_left_iff
thf(fact_1324_and__extract__pure__right__iff,axiom,
    ! [P: assn,B2: $o] :
      ( ( inf_inf_assn @ P @ ( pure_assn @ B2 ) )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ P ) @ ( pure_assn @ B2 ) ) ) ).

% and_extract_pure_right_iff
thf(fact_1325_order__refl,axiom,
    ! [X: assn] : ( ord_less_eq_assn @ X @ X ) ).

% order_refl
thf(fact_1326_order__refl,axiom,
    ! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).

% order_refl
thf(fact_1327_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_1328_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1329_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_1330_dual__order_Orefl,axiom,
    ! [A2: assn] : ( ord_less_eq_assn @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1331_dual__order_Orefl,axiom,
    ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1332_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1333_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1334_dual__order_Orefl,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1335_subsetI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A )
         => ( member8757157785044589968at_nat @ X3 @ B ) )
     => ( ord_le1268244103169919719at_nat @ A @ B ) ) ).

% subsetI
thf(fact_1336_subsetI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ( member8440522571783428010at_nat @ X3 @ B ) )
     => ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% subsetI
thf(fact_1337_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member_int @ X3 @ B ) )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% subsetI
thf(fact_1338_subsetI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A )
         => ( member6576561426505652726_nat_o @ X3 @ B ) )
     => ( ord_le2965882846123202637_nat_o @ A @ B ) ) ).

% subsetI
thf(fact_1339_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_1340_subset__antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_1341_inf_Oidem,axiom,
    ! [A2: assn] :
      ( ( inf_inf_assn @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_1342_inf_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_1343_inf_Oidem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_1344_inf__idem,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% inf_idem
thf(fact_1345_inf__idem,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_1346_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_1347_inf_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( inf_inf_assn @ A2 @ ( inf_inf_assn @ A2 @ B2 ) )
      = ( inf_inf_assn @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_1348_inf_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_1349_inf_Oleft__idem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
      = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_1350_inf__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_left_idem
thf(fact_1351_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_1352_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_1353_inf_Oright__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A2 @ B2 ) @ B2 )
      = ( inf_inf_assn @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_1354_inf_Oright__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_1355_inf_Oright__idem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ B2 )
      = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_1356_inf__right__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Y )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_right_idem
thf(fact_1357_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_1358_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_1359_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( ( ord_le3146513528884898305at_nat @ X @ Y )
        & ( ord_le3146513528884898305at_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1360_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( ( ord_less_eq_assn @ X @ Y )
        & ( ord_less_eq_assn @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1361_le__inf__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) )
      = ( ( ord_le2510731241096832064er_nat @ X @ Y )
        & ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1362_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1363_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1364_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1365_inf_Obounded__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) )
      = ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
        & ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1366_inf_Obounded__iff,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) )
      = ( ( ord_less_eq_assn @ A2 @ B2 )
        & ( ord_less_eq_assn @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1367_inf_Obounded__iff,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ ( inf_inf_filter_nat @ B2 @ C ) )
      = ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
        & ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1368_inf_Obounded__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
      = ( ( ord_less_eq_set_nat @ A2 @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1369_inf_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1370_inf_Obounded__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C ) )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1371_inf__bot__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% inf_bot_right
thf(fact_1372_inf__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% inf_bot_right
thf(fact_1373_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_1374_inf__bot__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_1375_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_1376_inf__bot__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_1377_inf__bot__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% inf_bot_left
thf(fact_1378_inf__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% inf_bot_left
thf(fact_1379_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_1380_inf__bot__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_1381_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_1382_inf__bot__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_1383_subset__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
      = ( A = bot_bo228742789529271731at_nat ) ) ).

% subset_empty
thf(fact_1384_subset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_1385_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1386_subset__empty,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1387_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1388_empty__subsetI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A ) ).

% empty_subsetI
thf(fact_1389_empty__subsetI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% empty_subsetI
thf(fact_1390_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_1391_empty__subsetI,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% empty_subsetI
thf(fact_1392_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_1393_insert__subset,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o2 @ X @ A ) @ B )
      = ( ( member_o @ X @ B )
        & ( ord_less_eq_set_o @ A @ B ) ) ) ).

% insert_subset
thf(fact_1394_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ B )
      = ( ( member8757157785044589968at_nat @ X @ B )
        & ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_1395_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ B )
      = ( ( member8440522571783428010at_nat @ X @ B )
        & ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_1396_insert__subset,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int2 @ X @ A ) @ B )
      = ( ( member_int @ X @ B )
        & ( ord_less_eq_set_int @ A @ B ) ) ) ).

% insert_subset
thf(fact_1397_insert__subset,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B )
      = ( ( member6576561426505652726_nat_o @ X @ B )
        & ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ).

% insert_subset
thf(fact_1398_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_1399_psubsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% psubsetI
thf(fact_1400_merge__pure__and,axiom,
    ! [A2: $o,B2: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_and
thf(fact_1401_singleton__insert__inj__eq,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1402_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1403_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A2: $o,A: set_o] :
      ( ( ( insert_o2 @ B2 @ bot_bot_set_o )
        = ( insert_o2 @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o2 @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1404_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( ( insert_int2 @ B2 @ bot_bot_set_int )
        = ( insert_int2 @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int2 @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1405_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat2 @ B2 @ bot_bot_set_nat )
        = ( insert_nat2 @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1406_singleton__insert__inj__eq_H,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ A )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
      = ( ( A2 = B2 )
        & ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1407_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ A )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( A2 = B2 )
        & ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1408_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B2: $o] :
      ( ( ( insert_o2 @ A2 @ A )
        = ( insert_o2 @ B2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o2 @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1409_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( ( insert_int2 @ A2 @ A )
        = ( insert_int2 @ B2 @ bot_bot_set_int ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int2 @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1410_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( ( insert_nat2 @ A2 @ A )
        = ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1411_Diff__eq__empty__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ord_le1268244103169919719at_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1412_Diff__eq__empty__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1413_Diff__eq__empty__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1414_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1415_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1416_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( ( inf_inf_assn @ P @ ( times_times_assn @ Q @ ( pure_assn @ B2 ) ) )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B2 ) ) ) ).

% and_extract_pure_right_ctx_iff
thf(fact_1417_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( ( inf_inf_assn @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B2 ) ) ) ).

% and_extract_pure_left_ctx_iff
thf(fact_1418_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_1419_in__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( member8757157785044589968at_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1420_in__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( member8440522571783428010at_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1421_in__mono,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( member_int @ X @ A )
       => ( member_int @ X @ B ) ) ) ).

% in_mono
thf(fact_1422_in__mono,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ B )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( member6576561426505652726_nat_o @ X @ B ) ) ) ).

% in_mono
thf(fact_1423_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1424_subsetD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( member8757157785044589968at_nat @ C @ A )
       => ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_1425_subsetD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( member8440522571783428010at_nat @ C @ A )
       => ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_1426_subsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% subsetD
thf(fact_1427_subsetD,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ B )
     => ( ( member6576561426505652726_nat_o @ C @ A )
       => ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% subsetD
thf(fact_1428_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_1429_equalityE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).

% equalityE
thf(fact_1430_pairwiseD,axiom,
    ! [R: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,S: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( pairwi3222497012029234898at_nat @ R @ S )
     => ( ( member8757157785044589968at_nat @ X @ S )
       => ( ( member8757157785044589968at_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_1431_pairwiseD,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( pairwi6357175311820755948at_nat @ R @ S )
     => ( ( member8440522571783428010at_nat @ X @ S )
       => ( ( member8440522571783428010at_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_1432_pairwiseD,axiom,
    ! [R: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
      ( ( pairwise_nat @ R @ S )
     => ( ( member_nat @ X @ S )
       => ( ( member_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_1433_pairwiseD,axiom,
    ! [R: int > int > $o,S: set_int,X: int,Y: int] :
      ( ( pairwise_int @ R @ S )
     => ( ( member_int @ X @ S )
       => ( ( member_int @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_1434_pairwiseD,axiom,
    ! [R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( pairwi2432143822833250616_nat_o @ R @ S )
     => ( ( member6576561426505652726_nat_o @ X @ S )
       => ( ( member6576561426505652726_nat_o @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_1435_pairwiseI,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
      ( ! [X3: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ S )
         => ( ( member8757157785044589968at_nat @ Y2 @ S )
           => ( ( X3 != Y2 )
             => ( R @ X3 @ Y2 ) ) ) )
     => ( pairwi3222497012029234898at_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_1436_pairwiseI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ S )
         => ( ( member8440522571783428010at_nat @ Y2 @ S )
           => ( ( X3 != Y2 )
             => ( R @ X3 @ Y2 ) ) ) )
     => ( pairwi6357175311820755948at_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_1437_pairwiseI,axiom,
    ! [S: set_nat,R: nat > nat > $o] :
      ( ! [X3: nat,Y2: nat] :
          ( ( member_nat @ X3 @ S )
         => ( ( member_nat @ Y2 @ S )
           => ( ( X3 != Y2 )
             => ( R @ X3 @ Y2 ) ) ) )
     => ( pairwise_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_1438_pairwiseI,axiom,
    ! [S: set_int,R: int > int > $o] :
      ( ! [X3: int,Y2: int] :
          ( ( member_int @ X3 @ S )
         => ( ( member_int @ Y2 @ S )
           => ( ( X3 != Y2 )
             => ( R @ X3 @ Y2 ) ) ) )
     => ( pairwise_int @ R @ S ) ) ).

% pairwiseI
thf(fact_1439_pairwiseI,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ S )
         => ( ( member6576561426505652726_nat_o @ Y2 @ S )
           => ( ( X3 != Y2 )
             => ( R @ X3 @ Y2 ) ) ) )
     => ( pairwi2432143822833250616_nat_o @ R @ S ) ) ).

% pairwiseI
thf(fact_1440_subset__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
        ! [X2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ A5 )
         => ( member8757157785044589968at_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1441_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A5 )
         => ( member8440522571783428010at_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1442_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B7: set_int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ A5 )
         => ( member_int @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1443_subset__eq,axiom,
    ( ord_le2965882846123202637_nat_o
    = ( ^ [A5: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
        ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X2 @ A5 )
         => ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1444_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ( member_nat @ X2 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1445_equalityD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% equalityD1
thf(fact_1446_equalityD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% equalityD2
thf(fact_1447_subset__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
        ! [T2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ T2 @ A5 )
         => ( member8757157785044589968at_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1448_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A5 )
         => ( member8440522571783428010at_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1449_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B7: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A5 )
         => ( member_int @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1450_subset__iff,axiom,
    ( ord_le2965882846123202637_nat_o
    = ( ^ [A5: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
        ! [T2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ T2 @ A5 )
         => ( member6576561426505652726_nat_o @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1451_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1452_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_1453_Collect__mono,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1454_Collect__mono,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q ) ) ) ).

% Collect_mono
thf(fact_1455_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_1456_subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_1457_pairwise__mono,axiom,
    ! [P: nat > nat > $o,A: set_nat,Q: nat > nat > $o,B: set_nat] :
      ( ( pairwise_nat @ P @ A )
     => ( ! [X3: nat,Y2: nat] :
            ( ( P @ X3 @ Y2 )
           => ( Q @ X3 @ Y2 ) )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( pairwise_nat @ Q @ B ) ) ) ) ).

% pairwise_mono
thf(fact_1458_set__eq__subset,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_1459_pairwise__subset,axiom,
    ! [P: nat > nat > $o,S: set_nat,T3: set_nat] :
      ( ( pairwise_nat @ P @ S )
     => ( ( ord_less_eq_set_nat @ T3 @ S )
       => ( pairwise_nat @ P @ T3 ) ) ) ).

% pairwise_subset
thf(fact_1460_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1461_Collect__mono__iff,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q ) )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1462_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1463_ord__eq__le__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn,D2: assn] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_assn @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1464_ord__eq__le__eq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat,D2: filter_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ C )
       => ( ( C = D2 )
         => ( ord_le2510731241096832064er_nat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1465_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat,D2: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_set_nat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1466_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_nat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1467_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_int @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1468_subset__Collect__conv,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ S @ ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ S )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1469_subset__Collect__conv,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ S @ ( collec939566748876313656_nat_o @ P ) )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ S )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1470_subset__Collect__conv,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ S )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1471_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% assn_aci(4)
thf(fact_1472_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% assn_aci(3)
thf(fact_1473_assn__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y5: assn] : ( inf_inf_assn @ Y5 @ X2 ) ) ) ).

% assn_aci(1)
thf(fact_1474_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(31)
thf(fact_1475_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(14)
thf(fact_1476_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_1477_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_1478_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1479_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1480_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: assn,Z2: assn] : Y4 = Z2 )
    = ( ^ [X2: assn,Y5: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y5 )
          & ( ord_less_eq_assn @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1481_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: filter_nat,Z2: filter_nat] : Y4 = Z2 )
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y5 )
          & ( ord_le2510731241096832064er_nat @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1482_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y5 )
          & ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1483_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [X2: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1484_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [X2: int,Y5: int] :
          ( ( ord_less_eq_int @ X2 @ Y5 )
          & ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1485_ord__eq__le__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ord_less_eq_assn @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1486_ord__eq__le__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( A2 = B2 )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ C )
       => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1487_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1488_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1489_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1490_ord__le__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_assn @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1491_ord__le__eq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1492_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1493_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1494_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1495_order__antisym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1496_order__antisym,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1497_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1498_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1499_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1500_order_Otrans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ord_less_eq_assn @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1501_order_Otrans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ C )
       => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1502_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1503_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1504_order_Otrans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1505_order__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_eq_assn @ X @ Z ) ) ) ).

% order_trans
thf(fact_1506_order__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
       => ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1507_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1508_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1509_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_1510_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1511_linorder__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int,B4: int] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1512_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: assn,Z2: assn] : Y4 = Z2 )
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ( ord_less_eq_assn @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1513_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: filter_nat,Z2: filter_nat] : Y4 = Z2 )
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1514_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1515_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1516_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1517_dual__order_Oantisym,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1518_dual__order_Oantisym,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1519_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1520_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1521_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1522_dual__order_Otrans,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C @ B2 )
       => ( ord_less_eq_assn @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1523_dual__order_Otrans,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ C @ B2 )
       => ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1524_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1525_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1526_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1527_antisym,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1528_antisym,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1529_antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1530_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1531_antisym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1532_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: assn,Z2: assn] : Y4 = Z2 )
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( ord_less_eq_assn @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1533_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: filter_nat,Z2: filter_nat] : Y4 = Z2 )
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1534_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1535_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1536_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1537_order__subst1,axiom,
    ! [A2: assn,F4: assn > assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1538_order__subst1,axiom,
    ! [A2: assn,F4: nat > assn,B2: nat,C: nat] :
      ( ( ord_less_eq_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1539_order__subst1,axiom,
    ! [A2: assn,F4: int > assn,B2: int,C: int] :
      ( ( ord_less_eq_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1540_order__subst1,axiom,
    ! [A2: nat,F4: assn > nat,B2: assn,C: assn] :
      ( ( ord_less_eq_nat @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1541_order__subst1,axiom,
    ! [A2: nat,F4: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1542_order__subst1,axiom,
    ! [A2: nat,F4: int > nat,B2: int,C: int] :
      ( ( ord_less_eq_nat @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1543_order__subst1,axiom,
    ! [A2: int,F4: assn > int,B2: assn,C: assn] :
      ( ( ord_less_eq_int @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1544_order__subst1,axiom,
    ! [A2: int,F4: nat > int,B2: nat,C: nat] :
      ( ( ord_less_eq_int @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1545_order__subst1,axiom,
    ! [A2: int,F4: int > int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1546_order__subst1,axiom,
    ! [A2: assn,F4: filter_nat > assn,B2: filter_nat,C: filter_nat] :
      ( ( ord_less_eq_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ C )
       => ( ! [X3: filter_nat,Y2: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1547_order__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1548_order__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > nat,C: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1549_order__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > int,C: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1550_order__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > assn,C: assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1551_order__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1552_order__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1553_order__subst2,axiom,
    ! [A2: int,B2: int,F4: int > assn,C: assn] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1554_order__subst2,axiom,
    ! [A2: int,B2: int,F4: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1555_order__subst2,axiom,
    ! [A2: int,B2: int,F4: int > int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1556_order__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > filter_nat,C: filter_nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_le2510731241096832064er_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_le2510731241096832064er_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le2510731241096832064er_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1557_order__eq__refl,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
     => ( ord_less_eq_assn @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1558_order__eq__refl,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
     => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1559_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1560_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1561_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1562_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1563_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_1564_ord__eq__le__subst,axiom,
    ! [A2: assn,F4: assn > assn,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1565_ord__eq__le__subst,axiom,
    ! [A2: nat,F4: assn > nat,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1566_ord__eq__le__subst,axiom,
    ! [A2: int,F4: assn > int,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1567_ord__eq__le__subst,axiom,
    ! [A2: assn,F4: nat > assn,B2: nat,C: nat] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1568_ord__eq__le__subst,axiom,
    ! [A2: nat,F4: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1569_ord__eq__le__subst,axiom,
    ! [A2: int,F4: nat > int,B2: nat,C: nat] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1570_ord__eq__le__subst,axiom,
    ! [A2: assn,F4: int > assn,B2: int,C: int] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1571_ord__eq__le__subst,axiom,
    ! [A2: nat,F4: int > nat,B2: int,C: int] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1572_ord__eq__le__subst,axiom,
    ! [A2: int,F4: int > int,B2: int,C: int] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1573_ord__eq__le__subst,axiom,
    ! [A2: filter_nat,F4: assn > filter_nat,B2: assn,C: assn] :
      ( ( A2
        = ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_le2510731241096832064er_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le2510731241096832064er_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1574_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1575_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > nat,C: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1576_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > int,C: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1577_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F4: nat > assn,C: assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1578_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F4: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1579_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F4: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1580_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F4: int > assn,C: assn] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1581_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F4: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1582_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F4: int > int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1583_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F4: assn > filter_nat,C: filter_nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F4 @ B2 )
          = C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_le2510731241096832064er_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le2510731241096832064er_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1584_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1585_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1586_order__antisym__conv,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1587_order__antisym__conv,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( ord_le2510731241096832064er_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1588_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1589_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1590_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1591_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_1592_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_1593_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_1594_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_1595_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_1596_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_1597_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_1598_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_1599_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_1600_inf__sup__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y5: assn] : ( inf_inf_assn @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_1601_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] : ( inf_inf_set_nat @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_1602_inf__sup__aci_I1_J,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_1603_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1604_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1605_inf__sup__ord_I2_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1606_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1607_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1608_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1609_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1610_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1611_inf__sup__ord_I1_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1612_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1613_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1614_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1615_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1616_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1617_inf__le1,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1618_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1619_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1620_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1621_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1622_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1623_inf__le2,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1624_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1625_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1626_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1627_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X @ A2 )
         => ~ ( ord_le3146513528884898305at_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1628_le__infE,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_assn @ X @ A2 )
         => ~ ( ord_less_eq_assn @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1629_le__infE,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ A2 @ B2 ) )
     => ~ ( ( ord_le2510731241096832064er_nat @ X @ A2 )
         => ~ ( ord_le2510731241096832064er_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1630_le__infE,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A2 )
         => ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1631_le__infE,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_nat @ X @ A2 )
         => ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1632_le__infE,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_int @ X @ A2 )
         => ~ ( ord_less_eq_int @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1633_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ X @ B2 )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1634_le__infI,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ A2 )
     => ( ( ord_less_eq_assn @ X @ B2 )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1635_le__infI,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ X @ B2 )
       => ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1636_le__infI,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ X @ B2 )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1637_le__infI,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ( ord_less_eq_nat @ X @ B2 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1638_le__infI,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ X @ B2 )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1639_inf__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ C )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( inf_in2572325071724192079at_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1640_inf__mono,axiom,
    ! [A2: assn,C: assn,B2: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C )
     => ( ( ord_less_eq_assn @ B2 @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ ( inf_inf_assn @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1641_inf__mono,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat,D2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ C )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ D2 )
       => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ ( inf_inf_filter_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1642_inf__mono,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ( ord_less_eq_set_nat @ B2 @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1643_inf__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1644_inf__mono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ ( inf_inf_int @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1645_le__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1646_le__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1647_le__infI1,axiom,
    ! [A2: filter_nat,X: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ X )
     => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1648_le__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1649_le__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1650_le__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1651_le__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B2 @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1652_le__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1653_le__infI2,axiom,
    ! [B2: filter_nat,X: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ X )
     => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1654_le__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1655_le__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1656_le__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1657_inf_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C )
      = ( inf_inf_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_1658_inf_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C )
      = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_1659_inf_Oassoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C )
      = ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_1660_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_1661_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_1662_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_1663_inf_OorderE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( A2
        = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1664_inf_OorderE,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( A2
        = ( inf_inf_assn @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1665_inf_OorderE,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_filter_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1666_inf_OorderE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1667_inf_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1668_inf_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( A2
        = ( inf_inf_int @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1669_inf_OorderI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1670_inf_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2
        = ( inf_inf_assn @ A2 @ B2 ) )
     => ( ord_less_eq_assn @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1671_inf_OorderI,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2
        = ( inf_inf_filter_nat @ A2 @ B2 ) )
     => ( ord_le2510731241096832064er_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1672_inf_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2
        = ( inf_inf_set_nat @ A2 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1673_inf_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( inf_inf_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1674_inf_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( inf_inf_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1675_inf__unique,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X3 @ Y2 )
             => ( ( ord_le3146513528884898305at_nat @ X3 @ Z4 )
               => ( ord_le3146513528884898305at_nat @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1676_inf__unique,axiom,
    ! [F4: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y2: assn] : ( ord_less_eq_assn @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: assn,Y2: assn] : ( ord_less_eq_assn @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: assn,Y2: assn,Z4: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ( ord_less_eq_assn @ X3 @ Z4 )
               => ( ord_less_eq_assn @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_assn @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1677_inf__unique,axiom,
    ! [F4: filter_nat > filter_nat > filter_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X3: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: filter_nat,Y2: filter_nat,Z4: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
             => ( ( ord_le2510731241096832064er_nat @ X3 @ Z4 )
               => ( ord_le2510731241096832064er_nat @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_filter_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1678_inf__unique,axiom,
    ! [F4: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: set_nat,Y2: set_nat,Z4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ( ord_less_eq_set_nat @ X3 @ Z4 )
               => ( ord_less_eq_set_nat @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_set_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1679_inf__unique,axiom,
    ! [F4: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: nat,Y2: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ( ord_less_eq_nat @ X3 @ Z4 )
               => ( ord_less_eq_nat @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1680_inf__unique,axiom,
    ! [F4: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F4 @ X3 @ Y2 ) @ X3 )
     => ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ ( F4 @ X3 @ Y2 ) @ Y2 )
       => ( ! [X3: int,Y2: int,Z4: int] :
              ( ( ord_less_eq_int @ X3 @ Y2 )
             => ( ( ord_less_eq_int @ X3 @ Z4 )
               => ( ord_less_eq_int @ X3 @ ( F4 @ Y2 @ Z4 ) ) ) )
         => ( ( inf_inf_int @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1681_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1682_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y5: assn] :
          ( ( inf_inf_assn @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1683_le__iff__inf,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( inf_inf_filter_nat @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1684_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( inf_inf_set_nat @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1685_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( inf_inf_nat @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1686_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y5: int] :
          ( ( inf_inf_int @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1687_inf_Oabsorb1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1688_inf_Oabsorb1,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1689_inf_Oabsorb1,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( inf_inf_filter_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1690_inf_Oabsorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1691_inf_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1692_inf_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1693_inf_Oabsorb2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1694_inf_Oabsorb2,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1695_inf_Oabsorb2,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( inf_inf_filter_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1696_inf_Oabsorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1697_inf_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1698_inf_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1699_inf_Ocommute,axiom,
    ( inf_inf_assn
    = ( ^ [A3: assn,B3: assn] : ( inf_inf_assn @ B3 @ A3 ) ) ) ).

% inf.commute
thf(fact_1700_inf_Ocommute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).

% inf.commute
thf(fact_1701_inf_Ocommute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B3 @ A3 ) ) ) ).

% inf.commute
thf(fact_1702_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1703_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1704_inf__absorb1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( inf_inf_filter_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1705_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( inf_inf_set_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1706_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1707_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1708_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y @ X )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1709_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1710_inf__absorb2,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( inf_inf_filter_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1711_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( inf_inf_set_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1712_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1713_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1714_inf__commute,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y5: assn] : ( inf_inf_assn @ Y5 @ X2 ) ) ) ).

% inf_commute
thf(fact_1715_inf__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] : ( inf_inf_set_nat @ Y5 @ X2 ) ) ) ).

% inf_commute
thf(fact_1716_inf__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y5 @ X2 ) ) ) ).

% inf_commute
thf(fact_1717_inf_OboundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
         => ~ ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1718_inf_OboundedE,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) )
     => ~ ( ( ord_less_eq_assn @ A2 @ B2 )
         => ~ ( ord_less_eq_assn @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1719_inf_OboundedE,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ ( inf_inf_filter_nat @ B2 @ C ) )
     => ~ ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
         => ~ ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1720_inf_OboundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1721_inf_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1722_inf_OboundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C ) )
     => ~ ( ( ord_less_eq_int @ A2 @ B2 )
         => ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_1723_inf_OboundedI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ C )
       => ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1724_inf_OboundedI,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ A2 @ C )
       => ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1725_inf_OboundedI,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( ord_le2510731241096832064er_nat @ A2 @ C )
       => ( ord_le2510731241096832064er_nat @ A2 @ ( inf_inf_filter_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1726_inf_OboundedI,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1727_inf_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1728_inf_OboundedI,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ A2 @ C )
       => ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1729_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X @ Z )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1730_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Z )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1731_inf__greatest,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ X @ Z )
       => ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1732_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1733_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1734_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1735_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( A3
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1736_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( A3
          = ( inf_inf_assn @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1737_inf_Oorder__iff,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( A3
          = ( inf_inf_filter_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1738_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( A3
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1739_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1740_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( A3
          = ( inf_inf_int @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1741_inf_Ocobounded1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1742_inf_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1743_inf_Ocobounded1,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1744_inf_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1745_inf_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1746_inf_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1747_inf_Ocobounded2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1748_inf_Ocobounded2,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1749_inf_Ocobounded2,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1750_inf_Ocobounded2,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1751_inf_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1752_inf_Ocobounded2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1753_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1754_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( inf_inf_assn @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1755_inf_Oabsorb__iff1,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( inf_inf_filter_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1756_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( inf_inf_set_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1757_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1758_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( inf_inf_int @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1759_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1760_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( inf_inf_assn @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1761_inf_Oabsorb__iff2,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( inf_inf_filter_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1762_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( inf_inf_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1763_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1764_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( inf_inf_int @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1765_inf_OcoboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1766_inf_OcoboundedI1,axiom,
    ! [A2: assn,C: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1767_inf_OcoboundedI1,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ C )
     => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1768_inf_OcoboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1769_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1770_inf_OcoboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1771_inf_OcoboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B2 @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1772_inf_OcoboundedI2,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1773_inf_OcoboundedI2,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ C )
     => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1774_inf_OcoboundedI2,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1775_inf_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1776_inf_OcoboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1777_inf_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( inf_inf_assn @ B2 @ ( inf_inf_assn @ A2 @ C ) )
      = ( inf_inf_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_1778_inf_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C ) )
      = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_1779_inf_Oleft__commute,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ B2 @ ( inf_in2572325071724192079at_nat @ A2 @ C ) )
      = ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_1780_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_1781_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_1782_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_1783_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1784_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B2 @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1785_inf_Ostrict__coboundedI2,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1786_inf_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1787_inf_Ostrict__coboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_int @ B2 @ C )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1788_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1789_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1790_inf_Ostrict__coboundedI1,axiom,
    ! [A2: assn,C: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1791_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1792_inf_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ A2 @ C )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1793_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( A3
            = ( inf_inf_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1794_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( A3
            = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1795_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( A3
            = ( inf_inf_assn @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1796_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( A3
            = ( inf_inf_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1797_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( A3
            = ( inf_inf_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1798_inf_Ostrict__boundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
     => ~ ( ( ord_less_set_nat @ A2 @ B2 )
         => ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1799_inf_Ostrict__boundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
         => ~ ( ord_le7866589430770878221at_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1800_inf_Ostrict__boundedE,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ ( inf_inf_assn @ B2 @ C ) )
     => ~ ( ( ord_less_assn @ A2 @ B2 )
         => ~ ( ord_less_assn @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1801_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
     => ~ ( ( ord_less_nat @ A2 @ B2 )
         => ~ ( ord_less_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1802_inf_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ ( inf_inf_int @ B2 @ C ) )
     => ~ ( ( ord_less_int @ A2 @ B2 )
         => ~ ( ord_less_int @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1803_inf_Oabsorb4,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_1804_inf_Oabsorb4,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B2 @ A2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_1805_inf_Oabsorb4,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_1806_inf_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_1807_inf_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_1808_inf_Oabsorb3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_1809_inf_Oabsorb3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_1810_inf_Oabsorb3,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_1811_inf_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_1812_inf_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_1813_less__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_1814_less__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B2 @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_1815_less__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_1816_less__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_1817_less__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( ( ord_less_int @ B2 @ X )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_1818_less__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_1819_less__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_1820_less__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_1821_less__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_1822_less__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_int @ A2 @ X )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_1823_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ~ ( ord_le3102999989581377725nteger @ X7 @ T ) ) ).

% pinf(6)
thf(fact_1824_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ~ ( ord_less_eq_num @ X7 @ T ) ) ).

% pinf(6)
thf(fact_1825_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ~ ( ord_less_eq_nat @ X7 @ T ) ) ).

% pinf(6)
thf(fact_1826_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ~ ( ord_less_eq_int @ X7 @ T ) ) ).

% pinf(6)
thf(fact_1827_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X7 )
     => ( ord_le3102999989581377725nteger @ T @ X7 ) ) ).

% pinf(8)
thf(fact_1828_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z4 @ X7 )
     => ( ord_less_eq_num @ T @ X7 ) ) ).

% pinf(8)
thf(fact_1829_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z4 @ X7 )
     => ( ord_less_eq_nat @ T @ X7 ) ) ).

% pinf(8)
thf(fact_1830_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z4 @ X7 )
     => ( ord_less_eq_int @ T @ X7 ) ) ).

% pinf(8)
thf(fact_1831_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ( ord_le3102999989581377725nteger @ X7 @ T ) ) ).

% minf(6)
thf(fact_1832_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ( ord_less_eq_num @ X7 @ T ) ) ).

% minf(6)
thf(fact_1833_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ( ord_less_eq_nat @ X7 @ T ) ) ).

% minf(6)
thf(fact_1834_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ( ord_less_eq_int @ X7 @ T ) ) ).

% minf(6)
thf(fact_1835_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z4: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z4 )
     => ~ ( ord_le3102999989581377725nteger @ T @ X7 ) ) ).

% minf(8)
thf(fact_1836_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z4 )
     => ~ ( ord_less_eq_num @ T @ X7 ) ) ).

% minf(8)
thf(fact_1837_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X7 ) ) ).

% minf(8)
thf(fact_1838_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X7 ) ) ).

% minf(8)
thf(fact_1839_order__le__imp__less__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1840_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1841_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1842_order__le__imp__less__or__eq,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_less_filter_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1843_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1844_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1845_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1846_linorder__le__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
      | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1847_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1848_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1849_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1850_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1851_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F4: code_integer > num,C: num] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1852_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > code_integer,C: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1853_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > num,C: num] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1854_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F4: num > code_integer,C: code_integer] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1855_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F4: num > num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1856_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > code_integer,C: code_integer] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1857_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > num,C: num] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1858_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F4: int > code_integer,C: code_integer] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1859_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F4: int > num,C: num] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1860_order__less__le__subst1,axiom,
    ! [A2: code_integer,F4: assn > code_integer,B2: assn,C: assn] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_le3102999989581377725nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1861_order__less__le__subst1,axiom,
    ! [A2: num,F4: assn > num,B2: assn,C: assn] :
      ( ( ord_less_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1862_order__less__le__subst1,axiom,
    ! [A2: assn,F4: assn > assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1863_order__less__le__subst1,axiom,
    ! [A2: nat,F4: assn > nat,B2: assn,C: assn] :
      ( ( ord_less_nat @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1864_order__less__le__subst1,axiom,
    ! [A2: int,F4: assn > int,B2: assn,C: assn] :
      ( ( ord_less_int @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1865_order__less__le__subst1,axiom,
    ! [A2: code_integer,F4: nat > code_integer,B2: nat,C: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3102999989581377725nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1866_order__less__le__subst1,axiom,
    ! [A2: num,F4: nat > num,B2: nat,C: nat] :
      ( ( ord_less_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1867_order__less__le__subst1,axiom,
    ! [A2: assn,F4: nat > assn,B2: nat,C: nat] :
      ( ( ord_less_assn @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1868_order__less__le__subst1,axiom,
    ! [A2: nat,F4: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1869_order__less__le__subst1,axiom,
    ! [A2: int,F4: nat > int,B2: nat,C: nat] :
      ( ( ord_less_int @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1870_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > code_integer,C: code_integer] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_le3102999989581377725nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1871_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > num,C: num] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1872_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1873_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > nat,C: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1874_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F4: assn > int,C: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1875_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > code_integer,C: code_integer] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3102999989581377725nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1876_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_num @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1877_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > assn,C: assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_assn @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_assn @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1878_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_nat @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1879_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F4: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F4 @ B2 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_int @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_int @ ( F4 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1880_order__le__less__subst1,axiom,
    ! [A2: code_integer,F4: code_integer > code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1881_order__le__less__subst1,axiom,
    ! [A2: num,F4: code_integer > num,B2: code_integer,C: code_integer] :
      ( ( ord_less_eq_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1882_order__le__less__subst1,axiom,
    ! [A2: code_integer,F4: assn > code_integer,B2: assn,C: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1883_order__le__less__subst1,axiom,
    ! [A2: num,F4: assn > num,B2: assn,C: assn] :
      ( ( ord_less_eq_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ! [X3: assn,Y2: assn] :
              ( ( ord_less_assn @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1884_order__le__less__subst1,axiom,
    ! [A2: code_integer,F4: num > code_integer,B2: num,C: num] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1885_order__le__less__subst1,axiom,
    ! [A2: num,F4: num > num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y2: num] :
              ( ( ord_less_num @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1886_order__le__less__subst1,axiom,
    ! [A2: code_integer,F4: nat > code_integer,B2: nat,C: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1887_order__le__less__subst1,axiom,
    ! [A2: num,F4: nat > num,B2: nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1888_order__le__less__subst1,axiom,
    ! [A2: code_integer,F4: int > code_integer,B2: int,C: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_le6747313008572928689nteger @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1889_order__le__less__subst1,axiom,
    ! [A2: num,F4: int > num,B2: int,C: int] :
      ( ( ord_less_eq_num @ A2 @ ( F4 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y2: int] :
              ( ( ord_less_int @ X3 @ Y2 )
             => ( ord_less_num @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) )
         => ( ord_less_num @ A2 @ ( F4 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1890_order__less__le__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1891_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1892_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1893_order__less__le__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_less_filter_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
       => ( ord_less_filter_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1894_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1895_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1896_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1897_order__le__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1898_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1899_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1900_order__le__less__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_less_filter_nat @ Y @ Z )
       => ( ord_less_filter_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1901_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1902_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1903_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1904_order__neq__le__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != B2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1905_order__neq__le__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_num @ A2 @ B2 )
       => ( ord_less_num @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1906_order__neq__le__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_assn @ A2 @ B2 )
       => ( ord_less_assn @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1907_order__neq__le__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 != B2 )
     => ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
       => ( ord_less_filter_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1908_order__neq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1909_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1910_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1911_order__le__neq__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1912_order__le__neq__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_num @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1913_order__le__neq__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_assn @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1914_order__le__neq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_filter_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1915_order__le__neq__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1916_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1917_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1918_order__less__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ord_le3102999989581377725nteger @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1919_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1920_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ord_less_eq_assn @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1921_order__less__imp__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_less_filter_nat @ X @ Y )
     => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1922_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1923_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1924_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1925_linorder__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
      = ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1926_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1927_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1928_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1929_linorder__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ X @ Y ) )
      = ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1930_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1931_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1932_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1933_order__less__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1934_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y5: num] :
          ( ( ord_less_eq_num @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1935_order__less__le,axiom,
    ( ord_less_assn
    = ( ^ [X2: assn,Y5: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1936_order__less__le,axiom,
    ( ord_less_filter_nat
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1937_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1938_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1939_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y5: int] :
          ( ( ord_less_eq_int @ X2 @ Y5 )
          & ( X2 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1940_order__le__less,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X2: code_integer,Y5: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1941_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X2: num,Y5: num] :
          ( ( ord_less_num @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1942_order__le__less,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y5: assn] :
          ( ( ord_less_assn @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1943_order__le__less,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( ord_less_filter_nat @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1944_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( ord_less_set_nat @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1945_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( ord_less_nat @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1946_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y5: int] :
          ( ( ord_less_int @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1947_dual__order_Ostrict__implies__order,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1948_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ord_less_eq_num @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1949_dual__order_Ostrict__implies__order,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ord_less_eq_assn @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1950_dual__order_Ostrict__implies__order,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ( ord_less_filter_nat @ B2 @ A2 )
     => ( ord_le2510731241096832064er_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1951_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1952_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1953_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1954_order_Ostrict__implies__order,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1955_order_Ostrict__implies__order,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ord_less_eq_num @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1956_order_Ostrict__implies__order,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ord_less_eq_assn @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1957_order_Ostrict__implies__order,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_less_filter_nat @ A2 @ B2 )
     => ( ord_le2510731241096832064er_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1958_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1959_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1960_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1961_dual__order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
          & ~ ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1962_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1963_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ~ ( ord_less_eq_assn @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1964_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_filter_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ~ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1965_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1966_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1967_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1968_dual__order_Ostrict__trans2,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C @ B2 )
       => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1969_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1970_dual__order_Ostrict__trans2,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C @ B2 )
       => ( ord_less_assn @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1971_dual__order_Ostrict__trans2,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ( ord_less_filter_nat @ B2 @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ C @ B2 )
       => ( ord_less_filter_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1972_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1973_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1974_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1975_dual__order_Ostrict__trans1,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C @ B2 )
       => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1976_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1977_dual__order_Ostrict__trans1,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_assn @ C @ B2 )
       => ( ord_less_assn @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1978_dual__order_Ostrict__trans1,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( ord_less_filter_nat @ C @ B2 )
       => ( ord_less_filter_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1979_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1980_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1981_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1982_dual__order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1983_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1984_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1985_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_filter_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1986_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1987_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1988_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1989_dual__order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1990_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_num @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1991_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_assn @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1992_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_less_filter_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1993_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1994_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1995_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1996_order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
          & ~ ( ord_le3102999989581377725nteger @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1997_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1998_order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ~ ( ord_less_eq_assn @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1999_order_Ostrict__iff__not,axiom,
    ( ord_less_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ~ ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2000_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2001_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2002_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2003_order_Ostrict__trans2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2004_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2005_order_Ostrict__trans2,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ C )
       => ( ord_less_assn @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2006_order_Ostrict__trans2,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_less_filter_nat @ A2 @ B2 )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ C )
       => ( ord_less_filter_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2007_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2008_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2009_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2010_order_Ostrict__trans1,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2011_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2012_order_Ostrict__trans1,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ B2 @ C )
       => ( ord_less_assn @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2013_order_Ostrict__trans1,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( ord_less_filter_nat @ B2 @ C )
       => ( ord_less_filter_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2014_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2015_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2016_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2017_order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2018_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2019_order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2020_order_Ostrict__iff__order,axiom,
    ( ord_less_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2021_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2022_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2023_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2024_order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2025_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2026_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_assn @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2027_order_Oorder__iff__strict,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_less_filter_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2028_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2029_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2030_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2031_not__le__imp__less,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2032_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2033_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2034_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2035_less__le__not__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y5 )
          & ~ ( ord_le3102999989581377725nteger @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2036_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y5: num] :
          ( ( ord_less_eq_num @ X2 @ Y5 )
          & ~ ( ord_less_eq_num @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2037_less__le__not__le,axiom,
    ( ord_less_assn
    = ( ^ [X2: assn,Y5: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y5 )
          & ~ ( ord_less_eq_assn @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2038_less__le__not__le,axiom,
    ( ord_less_filter_nat
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y5 )
          & ~ ( ord_le2510731241096832064er_nat @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2039_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y5 )
          & ~ ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2040_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2041_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y5: int] :
          ( ( ord_less_eq_int @ X2 @ Y5 )
          & ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2042_antisym__conv2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2043_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2044_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ~ ( ord_less_assn @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2045_antisym__conv2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2046_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2047_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2048_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2049_antisym__conv1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2050_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2051_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2052_antisym__conv1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ~ ( ord_less_filter_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2053_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2054_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2055_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2056_nless__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2057_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ( ~ ( ord_less_num @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2058_nless__le,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ~ ( ord_less_assn @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_assn @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2059_nless__le,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ~ ( ord_less_filter_nat @ A2 @ B2 ) )
      = ( ~ ( ord_le2510731241096832064er_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2060_nless__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2061_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2062_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2063_leI,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% leI
thf(fact_2064_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_2065_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_2066_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_2067_leD,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ~ ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% leD
thf(fact_2068_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_2069_leD,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ~ ( ord_less_assn @ X @ Y ) ) ).

% leD
thf(fact_2070_leD,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ~ ( ord_less_filter_nat @ X @ Y ) ) ).

% leD
thf(fact_2071_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_2072_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_2073_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_2074_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ bot_bo228742789529271731at_nat )
     => ( A2 = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2075_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2076_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_2077_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_2078_bot_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ A2 @ bot_bot_assn )
     => ( A2 = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
thf(fact_2079_bot_Oextremum__uniqueI,axiom,
    ! [A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
     => ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2080_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2081_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2082_bot_Oextremum__unique,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_unique
thf(fact_2083_bot_Oextremum__unique,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_2084_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_2085_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_2086_bot_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ A2 @ bot_bot_assn )
      = ( A2 = bot_bot_assn ) ) ).

% bot.extremum_unique
thf(fact_2087_bot_Oextremum__unique,axiom,
    ! [A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
      = ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
thf(fact_2088_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_2089_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_2090_bot_Oextremum,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A2 ) ).

% bot.extremum
thf(fact_2091_bot_Oextremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).

% bot.extremum
thf(fact_2092_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_2093_bot_Oextremum,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% bot.extremum
thf(fact_2094_bot_Oextremum,axiom,
    ! [A2: assn] : ( ord_less_eq_assn @ bot_bot_assn @ A2 ) ).

% bot.extremum
thf(fact_2095_bot_Oextremum,axiom,
    ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A2 ) ).

% bot.extremum
thf(fact_2096_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_2097_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_2098_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2099_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2100_subrelI,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
      ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ R2 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le653643898420964396nteger @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2101_subrelI,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S3: set_Pr3286484037609594932et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le5966269811547037844et_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2102_subrelI,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S3: set_Pr8536935166611901872et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le4763372923235995152et_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2103_subrelI,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ R2 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le8725513860283290265nt_int @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2104_subrelI,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ R2 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y2 ) @ S3 ) )
     => ( ord_le135402666524580259nt_int @ R2 @ S3 ) ) ).

% subrelI
thf(fact_2105_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
        = ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2106_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A2 @ B2 )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2107_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_2108_diff__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_2109_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_2110_diff__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_2111_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ D2 @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2112_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2113_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & ( rep_assn @ Q @ H2 ) ) ) ).

% mod_and_dist
thf(fact_2114_ent__conjI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entails @ A @ B )
     => ( ( entails @ A @ C2 )
       => ( entails @ A @ ( inf_inf_assn @ B @ C2 ) ) ) ) ).

% ent_conjI
thf(fact_2115_ent__conjE1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( entails @ A @ C2 )
     => ( entails @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% ent_conjE1
thf(fact_2116_ent__conjE2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( entails @ B @ C2 )
     => ( entails @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% ent_conjE2
thf(fact_2117_subset__insertI2,axiom,
    ! [A: set_int,B: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ ( insert_int2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_2118_subset__insertI2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_2119_subset__insertI2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_2120_subset__insertI2,axiom,
    ! [A: set_o,B: set_o,B2: $o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_2121_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_2122_subset__insertI,axiom,
    ! [B: set_int,A2: int] : ( ord_less_eq_set_int @ B @ ( insert_int2 @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2123_subset__insertI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B @ ( insert8211810215607154385at_nat @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2124_subset__insertI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( insert9069300056098147895at_nat @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2125_subset__insertI,axiom,
    ! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o2 @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2126_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2127_subset__insert,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( ord_less_eq_set_o @ A @ ( insert_o2 @ X @ B ) )
        = ( ord_less_eq_set_o @ A @ B ) ) ) ).

% subset_insert
thf(fact_2128_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A )
     => ( ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
        = ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_2129_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
        = ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_2130_subset__insert,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( ord_less_eq_set_int @ A @ ( insert_int2 @ X @ B ) )
        = ( ord_less_eq_set_int @ A @ B ) ) ) ).

% subset_insert
thf(fact_2131_subset__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A )
     => ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
        = ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ).

% subset_insert
thf(fact_2132_subset__insert,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_2133_insert__mono,axiom,
    ! [C2: set_int,D: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ C2 @ D )
     => ( ord_less_eq_set_int @ ( insert_int2 @ A2 @ C2 ) @ ( insert_int2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_2134_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A2 @ C2 ) @ ( insert8211810215607154385at_nat @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_2135_insert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ D )
     => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A2 @ C2 ) @ ( insert9069300056098147895at_nat @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_2136_insert__mono,axiom,
    ! [C2: set_o,D: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D )
     => ( ord_less_eq_set_o @ ( insert_o2 @ A2 @ C2 ) @ ( insert_o2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_2137_insert__mono,axiom,
    ! [C2: set_nat,D: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D )
     => ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_2138_subset__code_I1_J,axiom,
    ! [Xs: list_P5464809261938338413at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ B )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ ( set_Pr3765526544606949372at_nat @ Xs ) )
           => ( member8757157785044589968at_nat @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2139_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2140_subset__code_I1_J,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ ( set_Pr400311997697478754_nat_o @ Xs ) @ B )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ ( set_Pr400311997697478754_nat_o @ Xs ) )
           => ( member6576561426505652726_nat_o @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2141_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
           => ( member_int @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2142_subset__code_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn,B: set_Pr5949110396991348497n_assn] :
      ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ B )
      = ( ! [X2: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
           => ( member7957490590177025114n_assn @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2143_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2144_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_2145_Diff__subset,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_2146_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ D @ B )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_2147_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B7 )
          | ( A5 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2148_subset__psubset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_2149_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2150_psubset__subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_2151_psubset__imp__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_2152_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B7 )
          & ( A5 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_2153_psubsetE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ( ord_less_eq_set_nat @ B @ A ) ) ) ).

% psubsetE
thf(fact_2154_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% norm_assertion_simps(9)
thf(fact_2155_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% norm_assertion_simps(10)
thf(fact_2156_pairwise__empty,axiom,
    ! [P: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] : ( pairwi3222497012029234898at_nat @ P @ bot_bo228742789529271731at_nat ) ).

% pairwise_empty
thf(fact_2157_pairwise__empty,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o] : ( pairwi6357175311820755948at_nat @ P @ bot_bo2099793752762293965at_nat ) ).

% pairwise_empty
thf(fact_2158_pairwise__empty,axiom,
    ! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).

% pairwise_empty
thf(fact_2159_pairwise__empty,axiom,
    ! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).

% pairwise_empty
thf(fact_2160_pairwise__empty,axiom,
    ! [P: int > int > $o] : ( pairwise_int @ P @ bot_bot_set_int ) ).

% pairwise_empty
thf(fact_2161_pairwise__insert,axiom,
    ! [R2: $o > $o > $o,X: $o,S3: set_o] :
      ( ( pairwise_o @ R2 @ ( insert_o2 @ X @ S3 ) )
      = ( ! [Y5: $o] :
            ( ( ( member_o @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwise_o @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2162_pairwise__insert,axiom,
    ! [R2: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,X: produc3843707927480180839at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( pairwi3222497012029234898at_nat @ R2 @ ( insert9069300056098147895at_nat @ X @ S3 ) )
      = ( ! [Y5: produc3843707927480180839at_nat] :
            ( ( ( member8757157785044589968at_nat @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwi3222497012029234898at_nat @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2163_pairwise__insert,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( pairwi6357175311820755948at_nat @ R2 @ ( insert8211810215607154385at_nat @ X @ S3 ) )
      = ( ! [Y5: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwi6357175311820755948at_nat @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2164_pairwise__insert,axiom,
    ! [R2: nat > nat > $o,X: nat,S3: set_nat] :
      ( ( pairwise_nat @ R2 @ ( insert_nat2 @ X @ S3 ) )
      = ( ! [Y5: nat] :
            ( ( ( member_nat @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwise_nat @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2165_pairwise__insert,axiom,
    ! [R2: int > int > $o,X: int,S3: set_int] :
      ( ( pairwise_int @ R2 @ ( insert_int2 @ X @ S3 ) )
      = ( ! [Y5: int] :
            ( ( ( member_int @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwise_int @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2166_pairwise__insert,axiom,
    ! [R2: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,S3: set_Pr4532377907799695533_nat_o] :
      ( ( pairwi2432143822833250616_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ S3 ) )
      = ( ! [Y5: produc3658429121746597890et_nat > $o] :
            ( ( ( member6576561426505652726_nat_o @ Y5 @ S3 )
              & ( Y5 != X ) )
           => ( ( R2 @ X @ Y5 )
              & ( R2 @ Y5 @ X ) ) )
        & ( pairwi2432143822833250616_nat_o @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_2167_subset__singletonD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
     => ( ( A = bot_bo228742789529271731at_nat )
        | ( A
          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2168_subset__singletonD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A = bot_bo2099793752762293965at_nat )
        | ( A
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2169_subset__singletonD,axiom,
    ! [A: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o2 @ X @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_2170_subset__singletonD,axiom,
    ! [A: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int2 @ X @ bot_bot_set_int ) )
     => ( ( A = bot_bot_set_int )
        | ( A
          = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_2171_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2172_subset__singleton__iff,axiom,
    ! [X4: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X4 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
      = ( ( X4 = bot_bo228742789529271731at_nat )
        | ( X4
          = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2173_subset__singleton__iff,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X4 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( X4 = bot_bo2099793752762293965at_nat )
        | ( X4
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2174_subset__singleton__iff,axiom,
    ! [X4: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X4 @ ( insert_o2 @ A2 @ bot_bot_set_o ) )
      = ( ( X4 = bot_bot_set_o )
        | ( X4
          = ( insert_o2 @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_2175_subset__singleton__iff,axiom,
    ! [X4: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ X4 @ ( insert_int2 @ A2 @ bot_bot_set_int ) )
      = ( ( X4 = bot_bot_set_int )
        | ( X4
          = ( insert_int2 @ A2 @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_2176_subset__singleton__iff,axiom,
    ! [X4: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X4 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
      = ( ( X4 = bot_bot_set_nat )
        | ( X4
          = ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2177_subset__minus__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( minus_3314409938677909166at_nat @ A @ B )
        = bot_bo228742789529271731at_nat ) ) ).

% subset_minus_empty
thf(fact_2178_subset__minus__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( minus_1356011639430497352at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
thf(fact_2179_subset__minus__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_2180_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_2181_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_2182_subset__Diff__insert,axiom,
    ! [A: set_o,B: set_o,X: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o2 @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C2 ) )
        & ~ ( member_o @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2183_subset__Diff__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( minus_3314409938677909166at_nat @ B @ ( insert9069300056098147895at_nat @ X @ C2 ) ) )
      = ( ( ord_le1268244103169919719at_nat @ A @ ( minus_3314409938677909166at_nat @ B @ C2 ) )
        & ~ ( member8757157785044589968at_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2184_subset__Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( minus_1356011639430497352at_nat @ B @ ( insert8211810215607154385at_nat @ X @ C2 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ ( minus_1356011639430497352at_nat @ B @ C2 ) )
        & ~ ( member8440522571783428010at_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2185_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,X: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ ( insert_int2 @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ C2 ) )
        & ~ ( member_int @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2186_subset__Diff__insert,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B @ ( insert5175938949040314269_nat_o @ X @ C2 ) ) )
      = ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B @ C2 ) )
        & ~ ( member6576561426505652726_nat_o @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2187_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
        & ~ ( member_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_2188_set__update__subsetI,axiom,
    ! [Xs: list_P5464809261938338413at_nat,A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,I: nat] :
      ( ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ A )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2189_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2190_set__update__subsetI,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,I: nat] :
      ( ( ord_le2965882846123202637_nat_o @ ( set_Pr400311997697478754_nat_o @ Xs ) @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ord_le2965882846123202637_nat_o @ ( set_Pr400311997697478754_nat_o @ ( list_u6943956310655620667_nat_o @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2191_set__update__subsetI,axiom,
    ! [Xs: list_int,A: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A )
     => ( ( member_int @ X @ A )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2192_set__update__subsetI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,A: set_Pr5949110396991348497n_assn,X: produc6575502325842934193n_assn,I: nat] :
      ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ A )
     => ( ( member7957490590177025114n_assn @ X @ A )
       => ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2193_set__update__subsetI,axiom,
    ! [Xs: list_c,A: set_c,X: c,I: nat] :
      ( ( ord_less_eq_set_c @ ( set_c2 @ Xs ) @ A )
     => ( ( member_c @ X @ A )
       => ( ord_less_eq_set_c @ ( set_c2 @ ( list_update_c @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2194_set__update__subsetI,axiom,
    ! [Xs: list_d,A: set_d,X: d,I: nat] :
      ( ( ord_less_eq_set_d @ ( set_d2 @ Xs ) @ A )
     => ( ( member_d @ X @ A )
       => ( ord_less_eq_set_d @ ( set_d2 @ ( list_update_d @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2195_set__update__subsetI,axiom,
    ! [Xs: list_nat,A: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
     => ( ( member_nat @ X @ A )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_2196_set__remove1__subset,axiom,
    ! [X: int,Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( remove1_int @ X @ Xs ) ) @ ( set_int2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_2197_set__remove1__subset,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ ( remove1670527618125605709n_assn @ X @ Xs ) ) @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ).

% set_remove1_subset
thf(fact_2198_set__remove1__subset,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_2199_subset__code_I2_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Ys: list_P5464809261938338413at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( coset_5307619257043544794at_nat @ Ys ) )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ ( set_Pr3765526544606949372at_nat @ Ys ) )
           => ~ ( member8757157785044589968at_nat @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2200_subset__code_I2_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( coset_6809516720700967924at_nat @ Ys ) )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2201_subset__code_I2_J,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,Ys: list_P7985473006766602707_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ ( coset_9017067850271954240_nat_o @ Ys ) )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ ( set_Pr400311997697478754_nat_o @ Ys ) )
           => ~ ( member6576561426505652726_nat_o @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2202_subset__code_I2_J,axiom,
    ! [A: set_int,Ys: list_int] :
      ( ( ord_less_eq_set_int @ A @ ( coset_int @ Ys ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Ys ) )
           => ~ ( member_int @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2203_subset__code_I2_J,axiom,
    ! [A: set_Pr5949110396991348497n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ord_le171416862856029873n_assn @ A @ ( coset_703488708621576356n_assn @ Ys ) )
      = ( ! [X2: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Ys ) )
           => ~ ( member7957490590177025114n_assn @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2204_subset__code_I2_J,axiom,
    ! [A: set_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_2205_pairwise__singleton,axiom,
    ! [P: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] : ( pairwi3222497012029234898at_nat @ P @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% pairwise_singleton
thf(fact_2206_pairwise__singleton,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat] : ( pairwi6357175311820755948at_nat @ P @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% pairwise_singleton
thf(fact_2207_pairwise__singleton,axiom,
    ! [P: $o > $o > $o,A: $o] : ( pairwise_o @ P @ ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% pairwise_singleton
thf(fact_2208_pairwise__singleton,axiom,
    ! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% pairwise_singleton
thf(fact_2209_pairwise__singleton,axiom,
    ! [P: int > int > $o,A: int] : ( pairwise_int @ P @ ( insert_int2 @ A @ bot_bot_set_int ) ) ).

% pairwise_singleton
thf(fact_2210_subset__insert__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ A )
         => ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ A )
         => ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2211_subset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
      = ( ( ( member8757157785044589968at_nat @ X @ A )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ A )
         => ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2212_subset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
      = ( ( ( member8440522571783428010at_nat @ X @ A )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A )
         => ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2213_subset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o2 @ X @ B ) )
      = ( ( ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B ) )
        & ( ~ ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2214_subset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int2 @ X @ B ) )
      = ( ( ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B ) )
        & ( ~ ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2215_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
      = ( ( ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
        & ( ~ ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_2216_Diff__single__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B )
     => ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_2217_Diff__single__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B )
     => ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_2218_Diff__single__insert,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o2 @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_2219_Diff__single__insert,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B )
     => ( ord_less_eq_set_int @ A @ ( insert_int2 @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_2220_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_2221_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ ( insert8211810215607154385at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2222_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X ) ) @ ( insert9069300056098147895at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2223_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ ( insert_o2 @ X @ ( set_o2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2224_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2225_set__update__subset__insert,axiom,
    ! [Xs: list_P8527749157015355191n_assn,I: nat,X: produc6575502325842934193n_assn] : ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs @ I @ X ) ) @ ( insert5290817439147925377n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2226_set__update__subset__insert,axiom,
    ! [Xs: list_c,I: nat,X: c] : ( ord_less_eq_set_c @ ( set_c2 @ ( list_update_c @ Xs @ I @ X ) ) @ ( insert_c @ X @ ( set_c2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2227_set__update__subset__insert,axiom,
    ! [Xs: list_d,I: nat,X: d] : ( ord_less_eq_set_d @ ( set_d2 @ ( list_update_d @ Xs @ I @ X ) ) @ ( insert_d @ X @ ( set_d2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2228_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_2229_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_right
thf(fact_2230_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_2231_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_2232_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_2233_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_2234_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_2235_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_left
thf(fact_2236_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_2237_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_2238_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_2239_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_2240_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_2241_diff__shunt__var,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ X @ Y )
        = bot_bo228742789529271731at_nat )
      = ( ord_le1268244103169919719at_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2242_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2243_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2244_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2245_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( minus_minus_assn @ X @ Y )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2246_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2247_length__product,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( size_s2756448709718749209od_c_c @ ( product_c_c @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_product
thf(fact_2248_length__product,axiom,
    ! [Xs: list_c,Ys: list_d] :
      ( ( size_s2827482749765095194od_c_d @ ( product_c_d @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_product
thf(fact_2249_length__product,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( size_s2954062820725854115_c_nat @ ( product_c_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2250_length__product,axiom,
    ! [Xs: list_c,Ys: list_int] :
      ( ( size_s463979675794127359_c_int @ ( product_c_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2251_length__product,axiom,
    ! [Xs: list_d,Ys: list_c] :
      ( ( size_s2120799709113762520od_d_c @ ( product_d_c @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_product
thf(fact_2252_length__product,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( size_s2191833749160108505od_d_d @ ( product_d_d @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_product
thf(fact_2253_length__product,axiom,
    ! [Xs: list_d,Ys: list_nat] :
      ( ( size_s8550281435913984804_d_nat @ ( product_d_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2254_length__product,axiom,
    ! [Xs: list_d,Ys: list_int] :
      ( ( size_s6060198290982258048_d_int @ ( product_d_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2255_length__product,axiom,
    ! [Xs: list_nat,Ys: list_c] :
      ( ( size_s385972143775086793_nat_c @ ( product_nat_c @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_product
thf(fact_2256_length__product,axiom,
    ! [Xs: list_nat,Ys: list_d] :
      ( ( size_s457006183821432778_nat_d @ ( product_nat_d @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_product
thf(fact_2257_less__imp__diff__less,axiom,
    ! [J: nat,K3: nat,N: nat] :
      ( ( ord_less_nat @ J @ K3 )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K3 ) ) ).

% less_imp_diff_less
thf(fact_2258_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_2259_insert__subsetI,axiom,
    ! [X: $o,A: set_o,X4: set_o] :
      ( ( member_o @ X @ A )
     => ( ( ord_less_eq_set_o @ X4 @ A )
       => ( ord_less_eq_set_o @ ( insert_o2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2260_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,X4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A )
     => ( ( ord_le1268244103169919719at_nat @ X4 @ A )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2261_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ X4 @ A )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2262_insert__subsetI,axiom,
    ! [X: int,A: set_int,X4: set_int] :
      ( ( member_int @ X @ A )
     => ( ( ord_less_eq_set_int @ X4 @ A )
       => ( ord_less_eq_set_int @ ( insert_int2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2263_insert__subsetI,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,X4: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ A )
     => ( ( ord_le2965882846123202637_nat_o @ X4 @ A )
       => ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2264_insert__subsetI,axiom,
    ! [X: nat,A: set_nat,X4: set_nat] :
      ( ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X4 @ A )
       => ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_2265_subset__emptyI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ X3 @ A )
     => ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).

% subset_emptyI
thf(fact_2266_subset__emptyI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X3 @ A )
     => ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_2267_subset__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A )
     => ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_2268_subset__emptyI,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A )
     => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_2269_subset__emptyI,axiom,
    ! [A: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A )
     => ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_2270_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_2271_complete__interval,axiom,
    ! [A2: nat,B2: nat,P: nat > $o] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B2 )
         => ? [C4: nat] :
              ( ( ord_less_eq_nat @ A2 @ C4 )
              & ( ord_less_eq_nat @ C4 @ B2 )
              & ! [X7: nat] :
                  ( ( ( ord_less_eq_nat @ A2 @ X7 )
                    & ( ord_less_nat @ X7 @ C4 ) )
                 => ( P @ X7 ) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A2 @ X3 )
                        & ( ord_less_nat @ X3 @ D4 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D4 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_2272_complete__interval,axiom,
    ! [A2: int,B2: int,P: int > $o] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B2 )
         => ? [C4: int] :
              ( ( ord_less_eq_int @ A2 @ C4 )
              & ( ord_less_eq_int @ C4 @ B2 )
              & ! [X7: int] :
                  ( ( ( ord_less_eq_int @ A2 @ X7 )
                    & ( ord_less_int @ X7 @ C4 ) )
                 => ( P @ X7 ) )
              & ! [D4: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A2 @ X3 )
                        & ( ord_less_int @ X3 @ D4 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_int @ D4 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_2273_verit__comp__simplify1_I3_J,axiom,
    ! [B6: code_integer,A7: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ B6 @ A7 ) )
      = ( ord_le6747313008572928689nteger @ A7 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2274_verit__comp__simplify1_I3_J,axiom,
    ! [B6: num,A7: num] :
      ( ( ~ ( ord_less_eq_num @ B6 @ A7 ) )
      = ( ord_less_num @ A7 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2275_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A7: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
      = ( ord_less_nat @ A7 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2276_verit__comp__simplify1_I3_J,axiom,
    ! [B6: int,A7: int] :
      ( ( ~ ( ord_less_eq_int @ B6 @ A7 ) )
      = ( ord_less_int @ A7 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2277_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( ( ord_le3146513528884898305at_nat @ C2 @ A )
        & ( ord_le3146513528884898305at_nat @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_2278_Int__subset__iff,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
      = ( ( ord_less_eq_set_nat @ C2 @ A )
        & ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_2279_Int__insert__left__if0,axiom,
    ! [A2: $o,C2: set_o,B: set_o] :
      ( ~ ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ B ) @ C2 )
        = ( inf_inf_set_o @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2280_Int__insert__left__if0,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A2 @ C2 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ B ) @ C2 )
        = ( inf_in7913087082777306421at_nat @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2281_Int__insert__left__if0,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( ~ ( member_int @ A2 @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ B ) @ C2 )
        = ( inf_inf_set_int @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2282_Int__insert__left__if0,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A2 @ C2 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B ) @ C2 )
        = ( inf_in1906310914598751387_nat_o @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2283_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
        = ( inf_inf_set_nat @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2284_Int__insert__left__if0,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_2285_Int__insert__left__if1,axiom,
    ! [A2: $o,C2: set_o,B: set_o] :
      ( ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ B ) @ C2 )
        = ( insert_o2 @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2286_Int__insert__left__if1,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ C2 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ B ) @ C2 )
        = ( insert9069300056098147895at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2287_Int__insert__left__if1,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( ( member_int @ A2 @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ B ) @ C2 )
        = ( insert_int2 @ A2 @ ( inf_inf_set_int @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2288_Int__insert__left__if1,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ C2 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B ) @ C2 )
        = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2289_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
        = ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2290_Int__insert__left__if1,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B ) @ C2 )
        = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_2291_insert__inter__insert,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ A ) @ ( insert_int2 @ A2 @ B ) )
      = ( insert_int2 @ A2 @ ( inf_inf_set_int @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_2292_insert__inter__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ A ) @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( insert9069300056098147895at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_2293_insert__inter__insert,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ A ) @ ( insert_o2 @ A2 @ B ) )
      = ( insert_o2 @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_2294_insert__inter__insert,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ ( insert_nat2 @ A2 @ B ) )
      = ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_2295_insert__inter__insert,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_2296_Int__insert__right__if0,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ( inf_inf_set_o @ A @ ( insert_o2 @ A2 @ B ) )
        = ( inf_inf_set_o @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2297_Int__insert__right__if0,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A2 @ A )
     => ( ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
        = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2298_Int__insert__right__if0,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ A2 @ A )
     => ( ( inf_inf_set_int @ A @ ( insert_int2 @ A2 @ B ) )
        = ( inf_inf_set_int @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2299_Int__insert__right__if0,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B ) )
        = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2300_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2301_Int__insert__right__if0,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
        = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_2302_Int__insert__right__if1,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( inf_inf_set_o @ A @ ( insert_o2 @ A2 @ B ) )
        = ( insert_o2 @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2303_Int__insert__right__if1,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ A )
     => ( ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
        = ( insert9069300056098147895at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2304_Int__insert__right__if1,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( inf_inf_set_int @ A @ ( insert_int2 @ A2 @ B ) )
        = ( insert_int2 @ A2 @ ( inf_inf_set_int @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2305_Int__insert__right__if1,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B ) )
        = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2306_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
        = ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2307_Int__insert__right__if1,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
        = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_2308_insert__disjoint_I1_J,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( ( inf_in1906310914598751387_nat_o @ A @ B )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% insert_disjoint(1)
thf(fact_2309_insert__disjoint_I1_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ A ) @ B )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( ( inf_in7913087082777306421at_nat @ A @ B )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_2310_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( ( inf_in2572325071724192079at_nat @ A @ B )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_2311_insert__disjoint_I1_J,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ A ) @ B )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B )
        & ( ( inf_inf_set_o @ A @ B )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_2312_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ B )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ A @ B )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_2313_insert__disjoint_I1_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ A ) @ B )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B )
        & ( ( inf_inf_set_int @ A @ B )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_2314_insert__disjoint_I2_J,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B ) )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2315_insert__disjoint_I2_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2316_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2317_insert__disjoint_I2_J,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o2 @ A2 @ A ) @ B ) )
      = ( ~ ( member_o @ A2 @ B )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2318_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ A ) @ B ) )
      = ( ~ ( member_nat @ A2 @ B )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2319_insert__disjoint_I2_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int2 @ A2 @ A ) @ B ) )
      = ( ~ ( member_int @ A2 @ B )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_2320_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ A ) )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( ( inf_in1906310914598751387_nat_o @ B @ A )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% disjoint_insert(1)
thf(fact_2321_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ B @ ( insert9069300056098147895at_nat @ A2 @ A ) )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( ( inf_in7913087082777306421at_nat @ B @ A )
          = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_2322_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B @ ( insert8211810215607154385at_nat @ A2 @ A ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( ( inf_in2572325071724192079at_nat @ B @ A )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_2323_disjoint__insert_I1_J,axiom,
    ! [B: set_o,A2: $o,A: set_o] :
      ( ( ( inf_inf_set_o @ B @ ( insert_o2 @ A2 @ A ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B )
        & ( ( inf_inf_set_o @ B @ A )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_2324_disjoint__insert_I1_J,axiom,
    ! [B: set_nat,A2: nat,A: set_nat] :
      ( ( ( inf_inf_set_nat @ B @ ( insert_nat2 @ A2 @ A ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ B @ A )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_2325_disjoint__insert_I1_J,axiom,
    ! [B: set_int,A2: int,A: set_int] :
      ( ( ( inf_inf_set_int @ B @ ( insert_int2 @ A2 @ A ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B )
        & ( ( inf_inf_set_int @ B @ A )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_2326_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ B2 @ B ) ) )
      = ( ~ ( member6576561426505652726_nat_o @ B2 @ A )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2327_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ B ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B2 @ A )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2328_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B2 @ A )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2329_disjoint__insert_I2_J,axiom,
    ! [A: set_o,B2: $o,B: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A @ ( insert_o2 @ B2 @ B ) ) )
      = ( ~ ( member_o @ B2 @ A )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2330_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B2: nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) )
      = ( ~ ( member_nat @ B2 @ A )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2331_disjoint__insert_I2_J,axiom,
    ! [A: set_int,B2: int,B: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A @ ( insert_int2 @ B2 @ B ) ) )
      = ( ~ ( member_int @ B2 @ A )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_2332_Diff__disjoint,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A @ ( minus_3314409938677909166at_nat @ B @ A ) )
      = bot_bo228742789529271731at_nat ) ).

% Diff_disjoint
thf(fact_2333_Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( minus_1356011639430497352at_nat @ B @ A ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_2334_Diff__disjoint,axiom,
    ! [A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_2335_Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B @ A ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_2336_Diff__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_2337_list__update__beyond,axiom,
    ! [Xs: list_c,I: nat,X: c] :
      ( ( ord_less_eq_nat @ ( size_size_list_c @ Xs ) @ I )
     => ( ( list_update_c @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_2338_list__update__beyond,axiom,
    ! [Xs: list_d,I: nat,X: d] :
      ( ( ord_less_eq_nat @ ( size_size_list_d @ Xs ) @ I )
     => ( ( list_update_d @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_2339_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_2340_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_2341_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
          & ( M != N2 ) ) ) ) ).

% nat_less_le
thf(fact_2342_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_2343_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N2: nat] :
          ( ( ord_less_nat @ M @ N2 )
          | ( M = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_2344_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_2345_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_2346_less__mono__imp__le__mono,axiom,
    ! [F4: nat > nat,I: nat,J: nat] :
      ( ! [I5: nat,J3: nat] :
          ( ( ord_less_nat @ I5 @ J3 )
         => ( ord_less_nat @ ( F4 @ I5 ) @ ( F4 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F4 @ I ) @ ( F4 @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_2347_disjoint__iff__not__equal,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A )
           => ! [Y5: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y5 @ B )
               => ( X2 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_2348_disjoint__iff__not__equal,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
           => ! [Y5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ B )
               => ( X2 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_2349_disjoint__iff__not__equal,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A )
           => ! [Y5: $o] :
                ( ( member_o @ Y5 @ B )
               => ( X2 = ~ Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_2350_disjoint__iff__not__equal,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ! [Y5: nat] :
                ( ( member_nat @ Y5 @ B )
               => ( X2 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_2351_disjoint__iff__not__equal,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A )
           => ! [Y5: int] :
                ( ( member_int @ Y5 @ B )
               => ( X2 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_2352_Int__empty__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Int_empty_right
thf(fact_2353_Int__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_2354_Int__empty__right,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ A @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_2355_Int__empty__right,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_2356_Int__empty__right,axiom,
    ! [A: set_int] :
      ( ( inf_inf_set_int @ A @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_2357_Int__empty__left,axiom,
    ! [B: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ B )
      = bot_bo228742789529271731at_nat ) ).

% Int_empty_left
thf(fact_2358_Int__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_2359_Int__empty__left,axiom,
    ! [B: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_2360_Int__empty__left,axiom,
    ! [B: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_2361_Int__empty__left,axiom,
    ! [B: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_2362_disjoint__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ A )
           => ~ ( member6576561426505652726_nat_o @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2363_disjoint__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A )
           => ~ ( member8757157785044589968at_nat @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2364_disjoint__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
           => ~ ( member8440522571783428010at_nat @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2365_disjoint__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A )
           => ~ ( member_o @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2366_disjoint__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ~ ( member_nat @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2367_disjoint__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A )
           => ~ ( member_int @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_2368_Int__emptyI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
     => ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o ) ) ).

% Int_emptyI
thf(fact_2369_Int__emptyI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A )
         => ~ ( member8757157785044589968at_nat @ X3 @ B ) )
     => ( ( inf_in7913087082777306421at_nat @ A @ B )
        = bot_bo228742789529271731at_nat ) ) ).

% Int_emptyI
thf(fact_2370_Int__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ~ ( member8440522571783428010at_nat @ X3 @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_2371_Int__emptyI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ~ ( member_o @ X3 @ B ) )
     => ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_2372_Int__emptyI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ~ ( member_nat @ X3 @ B ) )
     => ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_2373_Int__emptyI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ~ ( member_int @ X3 @ B ) )
     => ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_2374_disjointI,axiom,
    ! [A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A2 )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B2 ) )
     => ( ( inf_in1906310914598751387_nat_o @ A2 @ B2 )
        = bot_bo7824918357723582233_nat_o ) ) ).

% disjointI
thf(fact_2375_disjointI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A2 )
         => ~ ( member8757157785044589968at_nat @ X3 @ B2 ) )
     => ( ( inf_in7913087082777306421at_nat @ A2 @ B2 )
        = bot_bo228742789529271731at_nat ) ) ).

% disjointI
thf(fact_2376_disjointI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ~ ( member8440522571783428010at_nat @ X3 @ B2 ) )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
thf(fact_2377_disjointI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A2 )
         => ~ ( member_o @ X3 @ B2 ) )
     => ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% disjointI
thf(fact_2378_disjointI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ~ ( member_nat @ X3 @ B2 ) )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% disjointI
thf(fact_2379_disjointI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ~ ( member_int @ X3 @ B2 ) )
     => ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% disjointI
thf(fact_2380_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ S4 )
     => ( ( ( inf_in2572325071724192079at_nat @ A @ S4 )
          = ( inf_in2572325071724192079at_nat @ B @ S4 ) )
       => ( ( inf_in2572325071724192079at_nat @ A @ S )
          = ( inf_in2572325071724192079at_nat @ B @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2381_inter__eq__subsetI,axiom,
    ! [S: set_nat,S4: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ S4 )
     => ( ( ( inf_inf_set_nat @ A @ S4 )
          = ( inf_inf_set_nat @ B @ S4 ) )
       => ( ( inf_inf_set_nat @ A @ S )
          = ( inf_inf_set_nat @ B @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2382_Int__Collect__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ A @ ( collec6321179662152712658at_nat @ P ) ) @ ( inf_in7913087082777306421at_nat @ B @ ( collec6321179662152712658at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2383_Int__Collect__mono,axiom,
    ! [A: set_int,B: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B @ ( collect_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2384_Int__Collect__mono,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ B )
     => ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B @ ( collec939566748876313656_nat_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2385_Int__Collect__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2386_Int__Collect__mono,axiom,
    ! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2387_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ A )
     => ( ( ord_le3146513528884898305at_nat @ C2 @ B )
       => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_2388_Int__greatest,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ B )
       => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_2389_Int__absorb2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_2390_Int__absorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_2391_Int__absorb1,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_2392_Int__absorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_2393_Int__lower2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_2394_Int__lower2,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_2395_Int__lower1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_2396_Int__lower1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_2397_Int__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_2398_Int__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_2399_Int__insert__left,axiom,
    ! [A2: $o,C2: set_o,B: set_o] :
      ( ( ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ B ) @ C2 )
          = ( insert_o2 @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) )
      & ( ~ ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o2 @ A2 @ B ) @ C2 )
          = ( inf_inf_set_o @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2400_Int__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A2 @ C2 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ B ) @ C2 )
          = ( insert9069300056098147895at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ B @ C2 ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A2 @ C2 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ B ) @ C2 )
          = ( inf_in7913087082777306421at_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2401_Int__insert__left,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( ( ( member_int @ A2 @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ B ) @ C2 )
          = ( insert_int2 @ A2 @ ( inf_inf_set_int @ B @ C2 ) ) ) )
      & ( ~ ( member_int @ A2 @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int2 @ A2 @ B ) @ C2 )
          = ( inf_inf_set_int @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2402_Int__insert__left,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A2 @ C2 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B ) @ C2 )
          = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B @ C2 ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A2 @ C2 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B ) @ C2 )
          = ( inf_in1906310914598751387_nat_o @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2403_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( ( ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
          = ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
          = ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2404_Int__insert__left,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A2 @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B ) @ C2 )
          = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B ) @ C2 )
          = ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_2405_Int__insert__right,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( ( member_o @ A2 @ A )
       => ( ( inf_inf_set_o @ A @ ( insert_o2 @ A2 @ B ) )
          = ( insert_o2 @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) )
      & ( ~ ( member_o @ A2 @ A )
       => ( ( inf_inf_set_o @ A @ ( insert_o2 @ A2 @ B ) )
          = ( inf_inf_set_o @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2406_Int__insert__right,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A2 @ A )
       => ( ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
          = ( insert9069300056098147895at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ A @ B ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A2 @ A )
       => ( ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
          = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2407_Int__insert__right,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( ( member_int @ A2 @ A )
       => ( ( inf_inf_set_int @ A @ ( insert_int2 @ A2 @ B ) )
          = ( insert_int2 @ A2 @ ( inf_inf_set_int @ A @ B ) ) ) )
      & ( ~ ( member_int @ A2 @ A )
       => ( ( inf_inf_set_int @ A @ ( insert_int2 @ A2 @ B ) )
          = ( inf_inf_set_int @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2408_Int__insert__right,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A2 @ A )
       => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B ) )
          = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
       => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B ) )
          = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2409_Int__insert__right,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
          = ( insert_nat2 @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
      & ( ~ ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2410_Int__insert__right,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A2 @ A )
       => ( ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
          = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A2 @ A )
       => ( ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
          = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_2411_Int__Diff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A @ ( minus_1356011639430497352at_nat @ B @ C2 ) ) ) ).

% Int_Diff
thf(fact_2412_Int__Diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
      = ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).

% Int_Diff
thf(fact_2413_Diff__Int2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A @ C2 ) @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A @ C2 ) @ B ) ) ).

% Diff_Int2
thf(fact_2414_Diff__Int2,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ B ) ) ).

% Diff_Int2
thf(fact_2415_Diff__Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% Diff_Diff_Int
thf(fact_2416_Diff__Diff__Int,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% Diff_Diff_Int
thf(fact_2417_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A ) @ ( inf_in2572325071724192079at_nat @ C2 @ B ) ) ) ).

% Diff_Int_distrib
thf(fact_2418_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A ) @ ( inf_inf_set_nat @ C2 @ B ) ) ) ).

% Diff_Int_distrib
thf(fact_2419_Diff__Int__distrib2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ C2 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A @ C2 ) @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_2420_Diff__Int__distrib2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_2421_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ N )
           => ~ ( P @ N3 ) )
       => ( P @ N ) )
     => ? [N4: nat] :
          ( ( ord_less_eq_nat @ N4 @ N )
          & ( P @ N4 ) ) ) ).

% exists_leI
thf(fact_2422_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( A3
          = ( inf_inf_assn @ A3 @ B3 ) ) ) ) ).

% less_eq_assn_def
thf(fact_2423_less__diff__iff,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K3 @ M2 )
     => ( ( ord_less_eq_nat @ K3 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_2424_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_2425_disjoint__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A7: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ A7 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ B6 )
       => ( ( ( inf_in7913087082777306421at_nat @ A7 @ B6 )
            = bot_bo228742789529271731at_nat )
         => ( ( inf_in7913087082777306421at_nat @ A2 @ B2 )
            = bot_bo228742789529271731at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_2426_disjoint__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A7: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ A7 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ B6 )
       => ( ( ( inf_in2572325071724192079at_nat @ A7 @ B6 )
            = bot_bo2099793752762293965at_nat )
         => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
            = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_2427_disjoint__mono,axiom,
    ! [A2: set_o,A7: set_o,B2: set_o,B6: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ A7 )
     => ( ( ord_less_eq_set_o @ B2 @ B6 )
       => ( ( ( inf_inf_set_o @ A7 @ B6 )
            = bot_bot_set_o )
         => ( ( inf_inf_set_o @ A2 @ B2 )
            = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
thf(fact_2428_disjoint__mono,axiom,
    ! [A2: set_int,A7: set_int,B2: set_int,B6: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ A7 )
     => ( ( ord_less_eq_set_int @ B2 @ B6 )
       => ( ( ( inf_inf_set_int @ A7 @ B6 )
            = bot_bot_set_int )
         => ( ( inf_inf_set_int @ A2 @ B2 )
            = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
thf(fact_2429_disjoint__mono,axiom,
    ! [A2: set_nat,A7: set_nat,B2: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ A7 )
     => ( ( ord_less_eq_set_nat @ B2 @ B6 )
       => ( ( ( inf_inf_set_nat @ A7 @ B6 )
            = bot_bot_set_nat )
         => ( ( inf_inf_set_nat @ A2 @ B2 )
            = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
thf(fact_2430_disjoint__alt__simp2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ A @ B )
       != A )
      = ( ( inf_in7913087082777306421at_nat @ A @ B )
       != bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2431_disjoint__alt__simp2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
       != A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2432_disjoint__alt__simp2,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
       != A )
      = ( ( inf_inf_set_o @ A @ B )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
thf(fact_2433_disjoint__alt__simp2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
       != A )
      = ( ( inf_inf_set_int @ A @ B )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
thf(fact_2434_disjoint__alt__simp2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
       != A )
      = ( ( inf_inf_set_nat @ A @ B )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2435_disjoint__alt__simp1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ A @ B )
        = A )
      = ( ( inf_in7913087082777306421at_nat @ A @ B )
        = bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2436_disjoint__alt__simp1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
        = A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2437_disjoint__alt__simp1,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
        = A )
      = ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
thf(fact_2438_disjoint__alt__simp1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
        = A )
      = ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
thf(fact_2439_disjoint__alt__simp1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
        = A )
      = ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2440_Int__Diff__disjoint,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ A @ B ) )
      = bot_bo228742789529271731at_nat ) ).

% Int_Diff_disjoint
thf(fact_2441_Int__Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_2442_Int__Diff__disjoint,axiom,
    ! [A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ B ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_2443_Int__Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B ) @ ( minus_minus_set_int @ A @ B ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_2444_Int__Diff__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_2445_Diff__triv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
     => ( ( minus_3314409938677909166at_nat @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2446_Diff__triv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2447_Diff__triv,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2448_Diff__triv,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2449_Diff__triv,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2450_foldl__absorb1,axiom,
    ! [X: code_integer,Zs: list_Code_integer] :
      ( ( times_3573771949741848930nteger @ X @ ( foldl_1542389171486570077nteger @ times_3573771949741848930nteger @ one_one_Code_integer @ Zs ) )
      = ( foldl_1542389171486570077nteger @ times_3573771949741848930nteger @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_2451_foldl__absorb1,axiom,
    ! [X: assn,Zs: list_assn] :
      ( ( times_times_assn @ X @ ( foldl_assn_assn @ times_times_assn @ one_one_assn @ Zs ) )
      = ( foldl_assn_assn @ times_times_assn @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_2452_foldl__absorb1,axiom,
    ! [X: rat,Zs: list_rat] :
      ( ( times_times_rat @ X @ ( foldl_rat_rat @ times_times_rat @ one_one_rat @ Zs ) )
      = ( foldl_rat_rat @ times_times_rat @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_2453_foldl__absorb1,axiom,
    ! [X: nat,Zs: list_nat] :
      ( ( times_times_nat @ X @ ( foldl_nat_nat @ times_times_nat @ one_one_nat @ Zs ) )
      = ( foldl_nat_nat @ times_times_nat @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_2454_foldl__absorb1,axiom,
    ! [X: int,Zs: list_int] :
      ( ( times_times_int @ X @ ( foldl_int_int @ times_times_int @ one_one_int @ Zs ) )
      = ( foldl_int_int @ times_times_int @ X @ Zs ) ) ).

% foldl_absorb1
thf(fact_2455_length__removeAll__less__eq,axiom,
    ! [X: c,Xs: list_c] : ( ord_less_eq_nat @ ( size_size_list_c @ ( removeAll_c @ X @ Xs ) ) @ ( size_size_list_c @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_2456_length__removeAll__less__eq,axiom,
    ! [X: d,Xs: list_d] : ( ord_less_eq_nat @ ( size_size_list_d @ ( removeAll_d @ X @ Xs ) ) @ ( size_size_list_d @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_2457_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_2458_length__removeAll__less__eq,axiom,
    ! [X: int,Xs: list_int] : ( ord_less_eq_nat @ ( size_size_list_int @ ( removeAll_int @ X @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_2459_disjoint__alt__simp3,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ A )
      = ( ( inf_in7913087082777306421at_nat @ A @ B )
       != bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_2460_disjoint__alt__simp3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_2461_disjoint__alt__simp3,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_set_o @ ( minus_minus_set_o @ A @ B ) @ A )
      = ( ( inf_inf_set_o @ A @ B )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
thf(fact_2462_disjoint__alt__simp3,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ ( minus_minus_set_int @ A @ B ) @ A )
      = ( ( inf_inf_set_int @ A @ B )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
thf(fact_2463_disjoint__alt__simp3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A )
      = ( ( inf_inf_set_nat @ A @ B )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_2464_verit__comp__simplify1_I1_J,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_2465_verit__comp__simplify1_I1_J,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_2466_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_2467_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_2468_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_2469_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2470_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R: set_Pr1261947904930325089at_nat,S5: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2471_ssubst__Pair__rhs,axiom,
    ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S3: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S5: produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2472_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S3: produc3658429121746597890et_nat,R: set_Pr3286484037609594932et_nat,S5: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2473_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S3: produc3925858234332021118et_nat,R: set_Pr8536935166611901872et_nat,S5: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2474_ssubst__Pair__rhs,axiom,
    ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S5: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2475_ssubst__Pair__rhs,axiom,
    ! [R2: int > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S5: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2476_distinct__length__le,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( distin4809818325527169450n_assn @ Ys )
     => ( ( ( set_Pr1139785259514867910n_assn @ Ys )
          = ( set_Pr1139785259514867910n_assn @ Xs ) )
       => ( ord_less_eq_nat @ ( size_s6829681357464350627n_assn @ Ys ) @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_2477_distinct__length__le,axiom,
    ! [Ys: list_c,Xs: list_c] :
      ( ( distinct_c @ Ys )
     => ( ( ( set_c2 @ Ys )
          = ( set_c2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_c @ Ys ) @ ( size_size_list_c @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_2478_distinct__length__le,axiom,
    ! [Ys: list_d,Xs: list_d] :
      ( ( distinct_d @ Ys )
     => ( ( ( set_d2 @ Ys )
          = ( set_d2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_d @ Ys ) @ ( size_size_list_d @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_2479_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( distinct_nat @ Ys )
     => ( ( ( set_nat2 @ Ys )
          = ( set_nat2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_2480_distinct__length__le,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( distinct_int @ Ys )
     => ( ( ( set_int2 @ Ys )
          = ( set_int2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_2481_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_2482_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_2483_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_2484_less__not__refl3,axiom,
    ! [S3: nat,T: nat] :
      ( ( ord_less_nat @ S3 @ T )
     => ( S3 != T ) ) ).

% less_not_refl3
thf(fact_2485_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_2486_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N5: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N5 )
             => ( P @ M3 ) )
         => ( P @ N5 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_2487_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N5: nat] :
          ( ~ ( P @ N5 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N5 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_2488_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_2489_size__neq__size__imp__neq,axiom,
    ! [X: list_c,Y: list_c] :
      ( ( ( size_size_list_c @ X )
       != ( size_size_list_c @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_2490_size__neq__size__imp__neq,axiom,
    ! [X: list_d,Y: list_d] :
      ( ( ( size_size_list_d @ X )
       != ( size_size_list_d @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_2491_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_2492_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_2493_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( distinct_int @ Xs )
     => ( ( distinct_int @ Ys )
       => ( distin3744728255968310194nt_int @ ( product_int_int @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_2494_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( distinct_int @ Xs )
     => ( ( distinct_nat @ Ys )
       => ( distin7922579275477506902nt_nat @ ( product_int_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_2495_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( distinct_nat @ Xs )
     => ( ( distinct_int @ Ys )
       => ( distin2745374544067255638at_int @ ( product_nat_int @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_2496_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( distinct_nat @ Ys )
       => ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_2497_distinct__sorted__mono,axiom,
    ! [L: list_Code_integer,I: nat,J: nat] :
      ( ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ L )
     => ( ( distin1543349897113766820nteger @ L )
       => ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_s3445333598471063425nteger @ L ) )
           => ( ord_le6747313008572928689nteger @ ( nth_Code_integer @ L @ I ) @ ( nth_Code_integer @ L @ J ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_2498_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_2499_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_2500_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_2501_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_Code_integer,I: nat,J: nat] :
      ( ( distin1543349897113766820nteger @ L )
     => ( ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ L )
       => ( ( ord_less_nat @ I @ ( size_s3445333598471063425nteger @ L ) )
         => ( ( ord_less_nat @ J @ ( size_s3445333598471063425nteger @ L ) )
           => ( ( ord_le6747313008572928689nteger @ ( nth_Code_integer @ L @ I ) @ ( nth_Code_integer @ L @ J ) )
              = ( ord_less_nat @ I @ J ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_2502_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_2503_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_2504_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_2505_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_c] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_c @ Xs ) )
       => ( ( size_size_list_c @ ( slice_c @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_2506_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_d] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_d @ Xs ) )
       => ( ( size_size_list_d @ ( slice_d @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_2507_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( size_size_list_nat @ ( slice_nat @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_2508_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( size_size_list_int @ ( slice_int @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_2509_rev__update,axiom,
    ! [K3: nat,Xs: list_c,Y: c] :
      ( ( ord_less_nat @ K3 @ ( size_size_list_c @ Xs ) )
     => ( ( rev_c @ ( list_update_c @ Xs @ K3 @ Y ) )
        = ( list_update_c @ ( rev_c @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ K3 ) @ one_one_nat ) @ Y ) ) ) ).

% rev_update
thf(fact_2510_rev__update,axiom,
    ! [K3: nat,Xs: list_d,Y: d] :
      ( ( ord_less_nat @ K3 @ ( size_size_list_d @ Xs ) )
     => ( ( rev_d @ ( list_update_d @ Xs @ K3 @ Y ) )
        = ( list_update_d @ ( rev_d @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ K3 ) @ one_one_nat ) @ Y ) ) ) ).

% rev_update
thf(fact_2511_rev__update,axiom,
    ! [K3: nat,Xs: list_nat,Y: nat] :
      ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ Xs ) )
     => ( ( rev_nat @ ( list_update_nat @ Xs @ K3 @ Y ) )
        = ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K3 ) @ one_one_nat ) @ Y ) ) ) ).

% rev_update
thf(fact_2512_rev__update,axiom,
    ! [K3: nat,Xs: list_int,Y: int] :
      ( ( ord_less_nat @ K3 @ ( size_size_list_int @ Xs ) )
     => ( ( rev_int @ ( list_update_int @ Xs @ K3 @ Y ) )
        = ( list_update_int @ ( rev_int @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ K3 ) @ one_one_nat ) @ Y ) ) ) ).

% rev_update
thf(fact_2513_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_c,X: c] :
      ( ( ( K3
          = ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ one_one_nat ) )
       => ( ( butlast_c @ ( list_update_c @ Xs @ K3 @ X ) )
          = ( butlast_c @ Xs ) ) )
      & ( ( K3
         != ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ one_one_nat ) )
       => ( ( butlast_c @ ( list_update_c @ Xs @ K3 @ X ) )
          = ( list_update_c @ ( butlast_c @ Xs ) @ K3 @ X ) ) ) ) ).

% butlast_list_update
thf(fact_2514_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_d,X: d] :
      ( ( ( K3
          = ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ one_one_nat ) )
       => ( ( butlast_d @ ( list_update_d @ Xs @ K3 @ X ) )
          = ( butlast_d @ Xs ) ) )
      & ( ( K3
         != ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ one_one_nat ) )
       => ( ( butlast_d @ ( list_update_d @ Xs @ K3 @ X ) )
          = ( list_update_d @ ( butlast_d @ Xs ) @ K3 @ X ) ) ) ) ).

% butlast_list_update
thf(fact_2515_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_nat,X: nat] :
      ( ( ( K3
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs @ K3 @ X ) )
          = ( butlast_nat @ Xs ) ) )
      & ( ( K3
         != ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs @ K3 @ X ) )
          = ( list_update_nat @ ( butlast_nat @ Xs ) @ K3 @ X ) ) ) ) ).

% butlast_list_update
thf(fact_2516_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_int,X: int] :
      ( ( ( K3
          = ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) )
       => ( ( butlast_int @ ( list_update_int @ Xs @ K3 @ X ) )
          = ( butlast_int @ Xs ) ) )
      & ( ( K3
         != ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) )
       => ( ( butlast_int @ ( list_update_int @ Xs @ K3 @ X ) )
          = ( list_update_int @ ( butlast_int @ Xs ) @ K3 @ X ) ) ) ) ).

% butlast_list_update
thf(fact_2517_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_2518_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_2519_length__zip,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( size_s2756448709718749209od_c_c @ ( zip_c_c @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_zip
thf(fact_2520_length__zip,axiom,
    ! [Xs: list_c,Ys: list_d] :
      ( ( size_s2827482749765095194od_c_d @ ( zip_c_d @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_zip
thf(fact_2521_length__zip,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( size_s2954062820725854115_c_nat @ ( zip_c_nat @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_2522_length__zip,axiom,
    ! [Xs: list_c,Ys: list_int] :
      ( ( size_s463979675794127359_c_int @ ( zip_c_int @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_zip
thf(fact_2523_length__zip,axiom,
    ! [Xs: list_d,Ys: list_c] :
      ( ( size_s2120799709113762520od_d_c @ ( zip_d_c @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_zip
thf(fact_2524_length__zip,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( size_s2191833749160108505od_d_d @ ( zip_d_d @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_zip
thf(fact_2525_length__zip,axiom,
    ! [Xs: list_d,Ys: list_nat] :
      ( ( size_s8550281435913984804_d_nat @ ( zip_d_nat @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_zip
thf(fact_2526_length__zip,axiom,
    ! [Xs: list_d,Ys: list_int] :
      ( ( size_s6060198290982258048_d_int @ ( zip_d_int @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_zip
thf(fact_2527_length__zip,axiom,
    ! [Xs: list_nat,Ys: list_c] :
      ( ( size_s385972143775086793_nat_c @ ( zip_nat_c @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_c @ Ys ) ) ) ).

% length_zip
thf(fact_2528_length__zip,axiom,
    ! [Xs: list_nat,Ys: list_d] :
      ( ( size_s457006183821432778_nat_d @ ( zip_nat_d @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_d @ Ys ) ) ) ).

% length_zip
thf(fact_2529_star__assnI,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,As3: set_nat] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
       => ( ( ( inf_inf_set_nat @ As2 @ As3 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As2 @ As3 ) ) ) ) ) ) ).

% star_assnI
thf(fact_2530_mod__star__conv,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
            ( ( H2
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_2531_rotate1__length01,axiom,
    ! [Xs: list_c] :
      ( ( ord_less_eq_nat @ ( size_size_list_c @ Xs ) @ one_one_nat )
     => ( ( rotate1_c @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_2532_rotate1__length01,axiom,
    ! [Xs: list_d] :
      ( ( ord_less_eq_nat @ ( size_size_list_d @ Xs ) @ one_one_nat )
     => ( ( rotate1_d @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_2533_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_2534_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_2535_sup_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_2536_sup_Oidem,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_2537_sup_Oidem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_2538_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_2539_sup__idem,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% sup_idem
thf(fact_2540_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_2541_sup_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_2542_sup_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_2543_sup_Oleft__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
      = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_2544_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_2545_sup__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% sup_left_idem
thf(fact_2546_sup__left__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_2547_sup_Oright__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_2548_sup_Oright__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ B2 )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_2549_sup_Oright__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ B2 )
      = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_2550_IntI,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A )
     => ( ( member8757157785044589968at_nat @ C @ B )
       => ( member8757157785044589968at_nat @ C @ ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_2551_IntI,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ A )
     => ( ( member_int @ C @ B )
       => ( member_int @ C @ ( inf_inf_set_int @ A @ B ) ) ) ) ).

% IntI
thf(fact_2552_IntI,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ A )
     => ( ( member6576561426505652726_nat_o @ C @ B )
       => ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% IntI
thf(fact_2553_IntI,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A )
     => ( ( member_nat @ C @ B )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_2554_IntI,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A )
     => ( ( member8440522571783428010at_nat @ C @ B )
       => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_2555_Int__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( inf_in7913087082777306421at_nat @ A @ B ) )
      = ( ( member8757157785044589968at_nat @ C @ A )
        & ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% Int_iff
thf(fact_2556_Int__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A @ B ) )
      = ( ( member_int @ C @ A )
        & ( member_int @ C @ B ) ) ) ).

% Int_iff
thf(fact_2557_Int__iff,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B ) )
      = ( ( member6576561426505652726_nat_o @ C @ A )
        & ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% Int_iff
thf(fact_2558_Int__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
      = ( ( member_nat @ C @ A )
        & ( member_nat @ C @ B ) ) ) ).

% Int_iff
thf(fact_2559_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( ( member8440522571783428010at_nat @ C @ A )
        & ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% Int_iff
thf(fact_2560_UnCI,axiom,
    ! [C: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ C @ B )
       => ( member8440522571783428010at_nat @ C @ A ) )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_2561_UnCI,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( ( ~ ( member_int @ C @ B )
       => ( member_int @ C @ A ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnCI
thf(fact_2562_UnCI,axiom,
    ! [C: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
      ( ( ~ ( member6576561426505652726_nat_o @ C @ B )
       => ( member6576561426505652726_nat_o @ C @ A ) )
     => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B ) ) ) ).

% UnCI
thf(fact_2563_UnCI,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ A ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_2564_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C @ B )
       => ( member8757157785044589968at_nat @ C @ A ) )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_2565_Un__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A @ B ) )
      = ( ( member8440522571783428010at_nat @ C @ A )
        | ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_2566_Un__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A @ B ) )
      = ( ( member_int @ C @ A )
        | ( member_int @ C @ B ) ) ) ).

% Un_iff
thf(fact_2567_Un__iff,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B ) )
      = ( ( member6576561426505652726_nat_o @ C @ A )
        | ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% Un_iff
thf(fact_2568_Un__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
      = ( ( member_nat @ C @ A )
        | ( member_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_2569_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( ( member8757157785044589968at_nat @ C @ A )
        | ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_2570_min_Oidem,axiom,
    ! [A2: nat] :
      ( ( ord_min_nat @ A2 @ A2 )
      = A2 ) ).

% min.idem
thf(fact_2571_min_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_min_nat @ A2 @ ( ord_min_nat @ A2 @ B2 ) )
      = ( ord_min_nat @ A2 @ B2 ) ) ).

% min.left_idem
thf(fact_2572_min_Oright__idem,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 )
      = ( ord_min_nat @ A2 @ B2 ) ) ).

% min.right_idem
thf(fact_2573_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( ( ord_le1268244103169919719at_nat @ X @ Z )
        & ( ord_le1268244103169919719at_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2574_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2575_le__sup__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ X @ Y ) @ Z )
      = ( ( ord_le2510731241096832064er_nat @ X @ Z )
        & ( ord_le2510731241096832064er_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2576_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2577_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2578_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_2579_sup_Obounded__iff,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C ) @ A2 )
      = ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
        & ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2580_sup_Obounded__iff,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_assn @ B2 @ A2 )
        & ( ord_less_eq_assn @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2581_sup_Obounded__iff,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B2 @ C ) @ A2 )
      = ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
        & ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2582_sup_Obounded__iff,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_nat @ B2 @ A2 )
        & ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2583_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2584_sup_Obounded__iff,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_2585_sup__bot_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ bot_bot_assn )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2586_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2587_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2588_sup__bot_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2589_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2590_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_2591_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2592_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2593_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2594_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2595_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2596_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_2597_sup__bot_Oleft__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2598_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2599_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2600_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2601_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2602_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_2603_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( sup_sup_assn @ A2 @ B2 )
        = bot_bot_assn )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2604_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = bot_bo228742789529271731at_nat )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2605_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A2 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2606_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2607_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2608_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( sup_sup_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_2609_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = bot_bot_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_2610_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ X @ Y )
        = bot_bo228742789529271731at_nat )
      = ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_2611_sup__eq__bot__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_2612_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_2613_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_2614_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_2615_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X @ Y ) )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_2616_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_2617_bot__eq__sup__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_2618_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X @ Y ) )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_2619_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_2620_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X @ Y ) )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_2621_sup__bot__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% sup_bot_right
thf(fact_2622_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% sup_bot_right
thf(fact_2623_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% sup_bot_right
thf(fact_2624_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_2625_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_2626_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_2627_sup__bot__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% sup_bot_left
thf(fact_2628_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_2629_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_2630_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_2631_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_2632_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_2633_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2634_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2635_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2636_inf__sup__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2637_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2638_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2639_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2640_sup__inf__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2641_Un__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_2642_Un__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_2643_Un__empty,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 ) ) ) ).

% Un_empty
thf(fact_2644_Un__empty,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 ) ) ) ).

% Un_empty
thf(fact_2645_Un__empty,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 ) ) ) ).

% Un_empty
thf(fact_2646_min__arg__le_I2_J,axiom,
    ! [M2: assn,N: assn] :
      ( ( ord_less_eq_assn @ M2 @ ( ord_min_assn @ M2 @ N ) )
      = ( ( ord_min_assn @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2647_min__arg__le_I2_J,axiom,
    ! [M2: filter_nat,N: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ M2 @ ( ord_min_filter_nat @ M2 @ N ) )
      = ( ( ord_min_filter_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2648_min__arg__le_I2_J,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ ( ord_min_set_nat @ M2 @ N ) )
      = ( ( ord_min_set_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2649_min__arg__le_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( ord_min_nat @ M2 @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2650_min__arg__le_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( ord_min_int @ M2 @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2651_min__arg__le_I1_J,axiom,
    ! [N: assn,M2: assn] :
      ( ( ord_less_eq_assn @ N @ ( ord_min_assn @ M2 @ N ) )
      = ( ( ord_min_assn @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2652_min__arg__le_I1_J,axiom,
    ! [N: filter_nat,M2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ N @ ( ord_min_filter_nat @ M2 @ N ) )
      = ( ( ord_min_filter_nat @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2653_min__arg__le_I1_J,axiom,
    ! [N: set_nat,M2: set_nat] :
      ( ( ord_less_eq_set_nat @ N @ ( ord_min_set_nat @ M2 @ N ) )
      = ( ( ord_min_set_nat @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2654_min__arg__le_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ ( ord_min_nat @ M2 @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2655_min__arg__le_I1_J,axiom,
    ! [N: int,M2: int] :
      ( ( ord_less_eq_int @ N @ ( ord_min_int @ M2 @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2656_min__eq__arg_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_min_nat @ M2 @ N )
        = N )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_2657_min__eq__arg_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ( ord_min_int @ M2 @ N )
        = N )
      = ( ord_less_eq_int @ N @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_2658_min__eq__arg_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_min_nat @ M2 @ N )
        = M2 )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% min_eq_arg(1)
thf(fact_2659_min__eq__arg_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( ( ord_min_int @ M2 @ N )
        = M2 )
      = ( ord_less_eq_int @ M2 @ N ) ) ).

% min_eq_arg(1)
thf(fact_2660_min_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_2661_min_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_2662_min_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_2663_min_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_2664_min_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% min.bounded_iff
thf(fact_2665_min_Obounded__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C ) )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ A2 @ C ) ) ) ).

% min.bounded_iff
thf(fact_2666_Un__subset__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C2 )
      = ( ( ord_le1268244103169919719at_nat @ A @ C2 )
        & ( ord_le1268244103169919719at_nat @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_2667_Un__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A @ C2 )
        & ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_2668_Un__insert__right,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( sup_sup_set_int @ A @ ( insert_int2 @ A2 @ B ) )
      = ( insert_int2 @ A2 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_2669_Un__insert__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_2670_Un__insert__right,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( sup_sup_set_o @ A @ ( insert_o2 @ A2 @ B ) )
      = ( insert_o2 @ A2 @ ( sup_sup_set_o @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_2671_Un__insert__right,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( insert_nat2 @ A2 @ B ) )
      = ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_2672_Un__insert__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_2673_Un__insert__left,axiom,
    ! [A2: int,B: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ ( insert_int2 @ A2 @ B ) @ C2 )
      = ( insert_int2 @ A2 @ ( sup_sup_set_int @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_2674_Un__insert__left,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A2 @ B ) @ C2 )
      = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_2675_Un__insert__left,axiom,
    ! [A2: $o,B: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o2 @ A2 @ B ) @ C2 )
      = ( insert_o2 @ A2 @ ( sup_sup_set_o @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_2676_Un__insert__left,axiom,
    ! [A2: nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ B ) @ C2 )
      = ( insert_nat2 @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_2677_Un__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A2 @ B ) @ C2 )
      = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_2678_min__less__iff__conj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_min_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        & ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2679_min__less__iff__conj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_min_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        & ( ord_less_num @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2680_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        & ( ord_less_nat @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2681_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        & ( ord_less_int @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2682_min_Oabsorb4,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_2683_min_Oabsorb4,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_min_num @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_2684_min_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_2685_min_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_2686_min_Oabsorb3,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_2687_min_Oabsorb3,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_min_num @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_2688_min_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_2689_min_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_2690_min__arg__not__ge_I1_J,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_Code_integer @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2691_min__arg__not__ge_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ~ ( ord_less_num @ ( ord_min_num @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_num @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2692_min__arg__not__ge_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2693_min__arg__not__ge_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_int @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2694_min__arg__not__ge_I2_J,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ M2 @ N ) @ N ) )
      = ( ( ord_min_Code_integer @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2695_min__arg__not__ge_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ~ ( ord_less_num @ ( ord_min_num @ M2 @ N ) @ N ) )
      = ( ( ord_min_num @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2696_min__arg__not__ge_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N ) @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2697_min__arg__not__ge_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N ) @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2698_min__less__self__conv_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ A2 )
      = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% min_less_self_conv(1)
thf(fact_2699_min__less__self__conv_I1_J,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ ( ord_min_num @ A2 @ B2 ) @ A2 )
      = ( ord_less_num @ B2 @ A2 ) ) ).

% min_less_self_conv(1)
thf(fact_2700_min__less__self__conv_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 )
      = ( ord_less_nat @ B2 @ A2 ) ) ).

% min_less_self_conv(1)
thf(fact_2701_min__less__self__conv_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ A2 )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% min_less_self_conv(1)
thf(fact_2702_min__less__self__conv_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% min_less_self_conv(2)
thf(fact_2703_min__less__self__conv_I2_J,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ ( ord_min_num @ A2 @ B2 ) @ B2 )
      = ( ord_less_num @ A2 @ B2 ) ) ).

% min_less_self_conv(2)
thf(fact_2704_min__less__self__conv_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% min_less_self_conv(2)
thf(fact_2705_min__less__self__conv_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% min_less_self_conv(2)
thf(fact_2706_min__simps_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = A2 ) ) ).

% min_simps(1)
thf(fact_2707_min__simps_I1_J,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_min_assn @ A2 @ B2 )
        = A2 ) ) ).

% min_simps(1)
thf(fact_2708_min__simps_I1_J,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_min_num @ A2 @ B2 )
        = A2 ) ) ).

% min_simps(1)
thf(fact_2709_min__simps_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min_simps(1)
thf(fact_2710_min__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = A2 ) ) ).

% min_simps(1)
thf(fact_2711_min__simps_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = B2 ) ) ).

% min_simps(2)
thf(fact_2712_min__simps_I2_J,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( ord_min_assn @ A2 @ B2 )
        = B2 ) ) ).

% min_simps(2)
thf(fact_2713_min__simps_I2_J,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_min_num @ A2 @ B2 )
        = B2 ) ) ).

% min_simps(2)
thf(fact_2714_min__simps_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min_simps(2)
thf(fact_2715_min__simps_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = B2 ) ) ).

% min_simps(2)
thf(fact_2716_min__bot2,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_mi6693596345299712430at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% min_bot2
thf(fact_2717_min__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot2
thf(fact_2718_min__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% min_bot2
thf(fact_2719_min__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% min_bot2
thf(fact_2720_min__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% min_bot2
thf(fact_2721_min__bot2,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ X @ bot_bot_nat )
      = bot_bot_nat ) ).

% min_bot2
thf(fact_2722_min__bot,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_mi6693596345299712430at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% min_bot
thf(fact_2723_min__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot
thf(fact_2724_min__bot,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% min_bot
thf(fact_2725_min__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% min_bot
thf(fact_2726_min__bot,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% min_bot
thf(fact_2727_min__bot,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ bot_bot_nat @ X )
      = bot_bot_nat ) ).

% min_bot
thf(fact_2728_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T3 @ ( inf_in2572325071724192079at_nat @ S @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_2729_Int__Un__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_2730_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ T3 @ ( inf_in7913087082777306421at_nat @ S @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_2731_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T3 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2732_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T3 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2733_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T3 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2734_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_2735_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_2736_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_2737_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T3 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2738_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2739_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T3 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2740_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T3 @ ( sup_su6327502436637775413at_nat @ S @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_2741_Un__Int__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_2742_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ T3 @ ( sup_su5525570899277871387at_nat @ S @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_2743_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T3 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2744_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T3 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2745_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T3 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2746_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_2747_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_2748_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_2749_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T3 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2750_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2751_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T3 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2752_Un__Diff__cancel2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B @ A ) @ A )
      = ( sup_su5525570899277871387at_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_2753_Un__Diff__cancel2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
      = ( sup_sup_set_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_2754_Un__Diff__cancel,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( minus_3314409938677909166at_nat @ B @ A ) )
      = ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_2755_Un__Diff__cancel,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_2756_set__rev,axiom,
    ! [Xs: list_int] :
      ( ( set_int2 @ ( rev_int @ Xs ) )
      = ( set_int2 @ Xs ) ) ).

% set_rev
thf(fact_2757_set__rev,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( rev_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_rev
thf(fact_2758_set__rev,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( rev_Pr4855572775806611735n_assn @ Xs ) )
      = ( set_Pr1139785259514867910n_assn @ Xs ) ) ).

% set_rev
thf(fact_2759_length__rev,axiom,
    ! [Xs: list_c] :
      ( ( size_size_list_c @ ( rev_c @ Xs ) )
      = ( size_size_list_c @ Xs ) ) ).

% length_rev
thf(fact_2760_length__rev,axiom,
    ! [Xs: list_d] :
      ( ( size_size_list_d @ ( rev_d @ Xs ) )
      = ( size_size_list_d @ Xs ) ) ).

% length_rev
thf(fact_2761_length__rev,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rev
thf(fact_2762_length__rev,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rev_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rev
thf(fact_2763_distinct__rev,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ ( rev_int @ Xs ) )
      = ( distinct_int @ Xs ) ) ).

% distinct_rev
thf(fact_2764_distinct__rev,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ ( rev_nat @ Xs ) )
      = ( distinct_nat @ Xs ) ) ).

% distinct_rev
thf(fact_2765_set__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( set_int2 @ ( rotate1_int @ Xs ) )
      = ( set_int2 @ Xs ) ) ).

% set_rotate1
thf(fact_2766_set__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( rotate1_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_rotate1
thf(fact_2767_set__rotate1,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( rotate328796349445179396n_assn @ Xs ) )
      = ( set_Pr1139785259514867910n_assn @ Xs ) ) ).

% set_rotate1
thf(fact_2768_length__rotate1,axiom,
    ! [Xs: list_c] :
      ( ( size_size_list_c @ ( rotate1_c @ Xs ) )
      = ( size_size_list_c @ Xs ) ) ).

% length_rotate1
thf(fact_2769_length__rotate1,axiom,
    ! [Xs: list_d] :
      ( ( size_size_list_d @ ( rotate1_d @ Xs ) )
      = ( size_size_list_d @ Xs ) ) ).

% length_rotate1
thf(fact_2770_length__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate1
thf(fact_2771_length__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rotate1_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rotate1
thf(fact_2772_distinct1__rotate,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ ( rotate1_int @ Xs ) )
      = ( distinct_int @ Xs ) ) ).

% distinct1_rotate
thf(fact_2773_distinct1__rotate,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ ( rotate1_nat @ Xs ) )
      = ( distinct_nat @ Xs ) ) ).

% distinct1_rotate
thf(fact_2774_list__ex__rev,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( list_ex_int @ P @ ( rev_int @ Xs ) )
      = ( list_ex_int @ P @ Xs ) ) ).

% list_ex_rev
thf(fact_2775_list__ex__rev,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( list_ex_nat @ P @ ( rev_nat @ Xs ) )
      = ( list_ex_nat @ P @ Xs ) ) ).

% list_ex_rev
thf(fact_2776_set__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( set_int2 @ ( union_int @ Xs @ Ys ) )
      = ( sup_sup_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ Ys ) ) ) ).

% set_union
thf(fact_2777_set__union,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( union_944050417204428971n_assn @ Xs @ Ys ) )
      = ( sup_su1735938885662425829n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( set_Pr1139785259514867910n_assn @ Ys ) ) ) ).

% set_union
thf(fact_2778_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_2779_set__union,axiom,
    ! [Xs: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( union_4462254032241401953at_nat @ Xs @ Ys ) )
      = ( sup_su5525570899277871387at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( set_Pr3765526544606949372at_nat @ Ys ) ) ) ).

% set_union
thf(fact_2780_length__butlast,axiom,
    ! [Xs: list_c] :
      ( ( size_size_list_c @ ( butlast_c @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_2781_length__butlast,axiom,
    ! [Xs: list_d] :
      ( ( size_size_list_d @ ( butlast_d @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_2782_length__butlast,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_2783_length__butlast,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( butlast_int @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_2784_IntE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( inf_in7913087082777306421at_nat @ A @ B ) )
     => ~ ( ( member8757157785044589968at_nat @ C @ A )
         => ~ ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% IntE
thf(fact_2785_IntE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A @ B ) )
     => ~ ( ( member_int @ C @ A )
         => ~ ( member_int @ C @ B ) ) ) ).

% IntE
thf(fact_2786_IntE,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B ) )
     => ~ ( ( member6576561426505652726_nat_o @ C @ A )
         => ~ ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% IntE
thf(fact_2787_IntE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( member_nat @ C @ A )
         => ~ ( member_nat @ C @ B ) ) ) ).

% IntE
thf(fact_2788_IntE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A )
         => ~ ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% IntE
thf(fact_2789_IntD1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( inf_in7913087082777306421at_nat @ A @ B ) )
     => ( member8757157785044589968at_nat @ C @ A ) ) ).

% IntD1
thf(fact_2790_IntD1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A @ B ) )
     => ( member_int @ C @ A ) ) ).

% IntD1
thf(fact_2791_IntD1,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B ) )
     => ( member6576561426505652726_nat_o @ C @ A ) ) ).

% IntD1
thf(fact_2792_IntD1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
     => ( member_nat @ C @ A ) ) ).

% IntD1
thf(fact_2793_IntD1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( member8440522571783428010at_nat @ C @ A ) ) ).

% IntD1
thf(fact_2794_IntD2,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( inf_in7913087082777306421at_nat @ A @ B ) )
     => ( member8757157785044589968at_nat @ C @ B ) ) ).

% IntD2
thf(fact_2795_IntD2,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A @ B ) )
     => ( member_int @ C @ B ) ) ).

% IntD2
thf(fact_2796_IntD2,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B ) )
     => ( member6576561426505652726_nat_o @ C @ B ) ) ).

% IntD2
thf(fact_2797_IntD2,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
     => ( member_nat @ C @ B ) ) ).

% IntD2
thf(fact_2798_IntD2,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( member8440522571783428010at_nat @ C @ B ) ) ).

% IntD2
thf(fact_2799_Int__assoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).

% Int_assoc
thf(fact_2800_Int__assoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ).

% Int_assoc
thf(fact_2801_Int__absorb,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_2802_Int__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_2803_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] : ( inf_inf_set_nat @ B7 @ A5 ) ) ) ).

% Int_commute
thf(fact_2804_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B7 @ A5 ) ) ) ).

% Int_commute
thf(fact_2805_Un__Int__crazy,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A @ B ) @ ( sup_su6327502436637775413at_nat @ B @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_2806_Un__Int__crazy,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_2807_Un__Int__crazy,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A @ B ) @ ( inf_in7913087082777306421at_nat @ B @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A ) )
      = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_2808_Int__Un__distrib,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( sup_su6327502436637775413at_nat @ B @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_2809_Int__Un__distrib,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_2810_Int__Un__distrib,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A @ B ) @ ( inf_in7913087082777306421at_nat @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_2811_Un__Int__distrib,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A @ B ) @ ( sup_su6327502436637775413at_nat @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_2812_Un__Int__distrib,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_2813_Un__Int__distrib,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( inf_in7913087082777306421at_nat @ B @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_2814_Int__Un__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B @ C2 ) @ A )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B @ A ) @ ( inf_in2572325071724192079at_nat @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_2815_Int__Un__distrib2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_2816_Int__Un__distrib2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B @ C2 ) @ A )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B @ A ) @ ( inf_in7913087082777306421at_nat @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_2817_Int__left__absorb,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% Int_left_absorb
thf(fact_2818_Int__left__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% Int_left_absorb
thf(fact_2819_Un__Int__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B @ C2 ) @ A )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B @ A ) @ ( sup_su6327502436637775413at_nat @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_2820_Un__Int__distrib2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_2821_Un__Int__distrib2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B @ C2 ) @ A )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B @ A ) @ ( sup_su5525570899277871387at_nat @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_2822_Int__left__commute,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_2823_Int__left__commute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ B @ ( inf_in2572325071724192079at_nat @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_2824_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% less_assn_def
thf(fact_2825_UnE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A @ B ) )
     => ( ~ ( member8440522571783428010at_nat @ C @ A )
       => ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% UnE
thf(fact_2826_UnE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A @ B ) )
     => ( ~ ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% UnE
thf(fact_2827_UnE,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B ) )
     => ( ~ ( member6576561426505652726_nat_o @ C @ A )
       => ( member6576561426505652726_nat_o @ C @ B ) ) ) ).

% UnE
thf(fact_2828_UnE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
     => ( ~ ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% UnE
thf(fact_2829_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) )
     => ( ~ ( member8757157785044589968at_nat @ C @ A )
       => ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% UnE
thf(fact_2830_UnI1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_2831_UnI1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ A )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnI1
thf(fact_2832_UnI1,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ A )
     => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B ) ) ) ).

% UnI1
thf(fact_2833_UnI1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_2834_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_2835_UnI2,axiom,
    ! [C: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ B )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_2836_UnI2,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( ( member_int @ C @ B )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnI2
thf(fact_2837_UnI2,axiom,
    ! [C: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ B )
     => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B ) ) ) ).

% UnI2
thf(fact_2838_UnI2,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( ( member_nat @ C @ B )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_2839_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ B )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_2840_bex__Un,axiom,
    ! [A: set_nat,B: set_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: nat] :
            ( ( member_nat @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_2841_bex__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ? [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_2842_ball__Un,axiom,
    ! [A: set_nat,B: set_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_2843_ball__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_2844_Un__assoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_2845_Un__assoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_2846_Un__absorb,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_2847_Un__absorb,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_2848_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A5 ) ) ) ).

% Un_commute
thf(fact_2849_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B7 @ A5 ) ) ) ).

% Un_commute
thf(fact_2850_Un__left__absorb,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_2851_Un__left__absorb,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_2852_Un__left__commute,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
      = ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_2853_Un__left__commute,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_2854_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_2855_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_2856_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_2857_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_2858_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_2859_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_2860_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_2861_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_2862_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_2863_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_2864_inf__sup__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y5: assn] : ( sup_sup_assn @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_2865_inf__sup__aci_I5_J,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y5 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_2866_min_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A2 @ B2 ) @ C )
      = ( ord_min_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ).

% min.assoc
thf(fact_2867_min_Ocommute,axiom,
    ( ord_min_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_min_nat @ B3 @ A3 ) ) ) ).

% min.commute
thf(fact_2868_min_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_min_nat @ B2 @ ( ord_min_nat @ A2 @ C ) )
      = ( ord_min_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ).

% min.left_commute
thf(fact_2869_sup_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_2870_sup_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ C )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_2871_sup_Oassoc,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C )
      = ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_2872_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_2873_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_2874_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_2875_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_2876_sup_Ocommute,axiom,
    ( sup_sup_assn
    = ( ^ [A3: assn,B3: assn] : ( sup_sup_assn @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_2877_sup_Ocommute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_2878_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).

% sup_commute
thf(fact_2879_sup__commute,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y5: assn] : ( sup_sup_assn @ Y5 @ X2 ) ) ) ).

% sup_commute
thf(fact_2880_sup__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y5 @ X2 ) ) ) ).

% sup_commute
thf(fact_2881_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_2882_sup_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( sup_sup_assn @ B2 @ ( sup_sup_assn @ A2 @ C ) )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_2883_sup_Oleft__commute,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ C ) )
      = ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_2884_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_2885_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_2886_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_2887_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( zip_c_c @ ( rev_c @ Xs ) @ ( rev_c @ Ys ) )
        = ( rev_Product_prod_c_c @ ( zip_c_c @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2888_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_d] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( zip_c_d @ ( rev_c @ Xs ) @ ( rev_d @ Ys ) )
        = ( rev_Product_prod_c_d @ ( zip_c_d @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2889_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_c_nat @ ( rev_c @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr3799340238495523993_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2890_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_int] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( zip_c_int @ ( rev_c @ Xs ) @ ( rev_int @ Ys ) )
        = ( rev_Pr8844861255841103093_c_int @ ( zip_c_int @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2891_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_c] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( zip_d_c @ ( rev_d @ Xs ) @ ( rev_c @ Ys ) )
        = ( rev_Product_prod_d_c @ ( zip_d_c @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2892_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( zip_d_d @ ( rev_d @ Xs ) @ ( rev_d @ Ys ) )
        = ( rev_Product_prod_d_d @ ( zip_d_d @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2893_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_nat] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_d_nat @ ( rev_d @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr5034784567451918490_d_nat @ ( zip_d_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2894_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_int] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( zip_d_int @ ( rev_d @ Xs ) @ ( rev_int @ Ys ) )
        = ( rev_Pr856933547942721782_d_int @ ( zip_d_int @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2895_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_c] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( zip_nat_c @ ( rev_nat @ Xs ) @ ( rev_c @ Ys ) )
        = ( rev_Pr4566615052912869567_nat_c @ ( zip_nat_c @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2896_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_d] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( zip_nat_d @ ( rev_nat @ Xs ) @ ( rev_d @ Ys ) )
        = ( rev_Pr4566615057216098368_nat_d @ ( zip_nat_d @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_2897_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_2898_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P5464809261938338413at_nat,P: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
      ( ! [X3: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Xs ) )
         => ( ( member8757157785044589968at_nat @ Y2 @ ( set_Pr3765526544606949372at_nat @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted8831241538929206612at_nat @ P @ Xs )
       => ( sorted8831241538929206612at_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2899_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( member8440522571783428010at_nat @ Y2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted5214655850825725294at_nat @ P @ Xs )
       => ( sorted5214655850825725294at_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2900_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ ( set_Pr400311997697478754_nat_o @ Xs ) )
         => ( ( member6576561426505652726_nat_o @ Y2 @ ( set_Pr400311997697478754_nat_o @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted9199660237762088378_nat_o @ P @ Xs )
       => ( sorted9199660237762088378_nat_o @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2901_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Q: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
      ( ! [X3: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
         => ( ( member7957490590177025114n_assn @ Y2 @ ( set_Pr1139785259514867910n_assn @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted3986126766855596574n_assn @ P @ Xs )
       => ( sorted3986126766855596574n_assn @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2902_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
      ( ! [X3: nat,Y2: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted_wrt_nat @ P @ Xs )
       => ( sorted_wrt_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2903_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_int,P: int > int > $o,Q: int > int > $o] :
      ( ! [X3: int,Y2: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ( member_int @ Y2 @ ( set_int2 @ Xs ) )
           => ( ( P @ X3 @ Y2 )
             => ( Q @ X3 @ Y2 ) ) ) )
     => ( ( sorted_wrt_int @ P @ Xs )
       => ( sorted_wrt_int @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_2904_min__def,axiom,
    ( ord_min_assn
    = ( ^ [A3: assn,B3: assn] : ( if_assn @ ( ord_less_eq_assn @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_2905_min__def,axiom,
    ( ord_min_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_2906_min__def,axiom,
    ( ord_min_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_2907_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_2908_min__def,axiom,
    ( ord_min_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_2909_min__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_min_assn @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_2910_min__absorb1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_min_filter_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_2911_min__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_min_set_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_2912_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_min_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_2913_min__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_min_int @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_2914_min__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_min_assn @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_2915_min__absorb2,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( ord_min_filter_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_2916_min__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_min_set_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_2917_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_min_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_2918_min__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_min_int @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_2919_min_Omono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).

% min.mono
thf(fact_2920_min_Omono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ ( ord_min_int @ C @ D2 ) ) ) ) ).

% min.mono
thf(fact_2921_min_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( ord_min_nat @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_2922_min_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( A2
        = ( ord_min_int @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_2923_min_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( ord_min_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_2924_min_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( ord_min_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_2925_min_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% min.boundedE
thf(fact_2926_min_OboundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C ) )
     => ~ ( ( ord_less_eq_int @ A2 @ B2 )
         => ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).

% min.boundedE
thf(fact_2927_min_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ) ).

% min.boundedI
thf(fact_2928_min_OboundedI,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ A2 @ C )
       => ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C ) ) ) ) ).

% min.boundedI
thf(fact_2929_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( ord_min_nat @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_2930_min_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( A3
          = ( ord_min_int @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_2931_min_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_2932_min_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_2933_min_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_2934_min_Ocobounded2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_2935_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_2936_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_min_int @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_2937_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_2938_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_min_int @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_2939_min_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI1
thf(fact_2940_min_OcoboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI1
thf(fact_2941_min_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI2
thf(fact_2942_min_OcoboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI2
thf(fact_2943_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        | ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_2944_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        | ( ord_less_eq_int @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_2945_min_Ostrict__coboundedI2,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ C )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_2946_min_Ostrict__coboundedI2,axiom,
    ! [B2: num,C: num,A2: num] :
      ( ( ord_less_num @ B2 @ C )
     => ( ord_less_num @ ( ord_min_num @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_2947_min_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_2948_min_Ostrict__coboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_int @ B2 @ C )
     => ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_2949_min_Ostrict__coboundedI1,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ C )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_2950_min_Ostrict__coboundedI1,axiom,
    ! [A2: num,C: num,B2: num] :
      ( ( ord_less_num @ A2 @ C )
     => ( ord_less_num @ ( ord_min_num @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_2951_min_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_2952_min_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ A2 @ C )
     => ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_2953_min_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( A3
            = ( ord_min_Code_integer @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_2954_min_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( A3
            = ( ord_min_num @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_2955_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( A3
            = ( ord_min_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_2956_min_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( A3
            = ( ord_min_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_2957_min_Ostrict__boundedE,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( ord_min_Code_integer @ B2 @ C ) )
     => ~ ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
         => ~ ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).

% min.strict_boundedE
thf(fact_2958_min_Ostrict__boundedE,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ ( ord_min_num @ B2 @ C ) )
     => ~ ( ( ord_less_num @ A2 @ B2 )
         => ~ ( ord_less_num @ A2 @ C ) ) ) ).

% min.strict_boundedE
thf(fact_2959_min_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
     => ~ ( ( ord_less_nat @ A2 @ B2 )
         => ~ ( ord_less_nat @ A2 @ C ) ) ) ).

% min.strict_boundedE
thf(fact_2960_min_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ ( ord_min_int @ B2 @ C ) )
     => ~ ( ( ord_less_int @ A2 @ B2 )
         => ~ ( ord_less_int @ A2 @ C ) ) ) ).

% min.strict_boundedE
thf(fact_2961_min__less__iff__disj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        | ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_2962_min__less__iff__disj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_min_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        | ( ord_less_num @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_2963_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        | ( ord_less_nat @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_2964_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        | ( ord_less_int @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_2965_min__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_2966_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_2967_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2968_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2969_inf__sup__ord_I4_J,axiom,
    ! [Y: filter_nat,X: filter_nat] : ( ord_le2510731241096832064er_nat @ Y @ ( sup_sup_filter_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2970_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2971_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2972_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2973_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2974_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2975_inf__sup__ord_I3_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2976_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2977_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2978_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2979_le__supE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le1268244103169919719at_nat @ A2 @ X )
         => ~ ( ord_le1268244103169919719at_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2980_le__supE,axiom,
    ! [A2: assn,B2: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_assn @ A2 @ X )
         => ~ ( ord_less_eq_assn @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2981_le__supE,axiom,
    ! [A2: filter_nat,B2: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le2510731241096832064er_nat @ A2 @ X )
         => ~ ( ord_le2510731241096832064er_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2982_le__supE,axiom,
    ! [A2: set_nat,B2: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ X )
         => ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2983_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2984_le__supE,axiom,
    ! [A2: int,B2: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_int @ A2 @ X )
         => ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2985_le__supI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ X )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2986_le__supI,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ X )
     => ( ( ord_less_eq_assn @ B2 @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2987_le__supI,axiom,
    ! [A2: filter_nat,X: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ X )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ X )
       => ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2988_le__supI,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X )
     => ( ( ord_less_eq_set_nat @ B2 @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2989_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2990_le__supI,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ X )
     => ( ( ord_less_eq_int @ B2 @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2991_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2992_sup__ge1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge1
thf(fact_2993_sup__ge1,axiom,
    ! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2994_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2995_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2996_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_2997_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2998_sup__ge2,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge2
thf(fact_2999_sup__ge2,axiom,
    ! [Y: filter_nat,X: filter_nat] : ( ord_le2510731241096832064er_nat @ Y @ ( sup_sup_filter_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_3000_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_3001_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_3002_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_3003_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ A2 )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3004_le__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ A2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3005_le__supI1,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ A2 )
     => ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3006_le__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3007_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3008_le__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_3009_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ B2 )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3010_le__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ X @ B2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3011_le__supI2,axiom,
    ! [X: filter_nat,B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ B2 )
     => ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3012_le__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3013_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3014_le__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ X @ B2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_3015_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ D2 @ B2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D2 ) @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3016_sup_Omono,axiom,
    ! [C: assn,A2: assn,D2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C @ A2 )
     => ( ( ord_less_eq_assn @ D2 @ B2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C @ D2 ) @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3017_sup_Omono,axiom,
    ! [C: filter_nat,A2: filter_nat,D2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ C @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ D2 @ B2 )
       => ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ C @ D2 ) @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3018_sup_Omono,axiom,
    ! [C: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ( ord_less_eq_set_nat @ D2 @ B2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3019_sup_Omono,axiom,
    ! [C: nat,A2: nat,D2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D2 @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3020_sup_Omono,axiom,
    ! [C: int,A2: int,D2: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ( ord_less_eq_int @ D2 @ B2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_3021_sup__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ C )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3022_sup__mono,axiom,
    ! [A2: assn,C: assn,B2: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C )
     => ( ( ord_less_eq_assn @ B2 @ D2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ ( sup_sup_assn @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3023_sup__mono,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat,D2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ C )
     => ( ( ord_le2510731241096832064er_nat @ B2 @ D2 )
       => ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A2 @ B2 ) @ ( sup_sup_filter_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3024_sup__mono,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ( ord_less_eq_set_nat @ B2 @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3025_sup__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3026_sup__mono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_3027_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X )
     => ( ( ord_le1268244103169919719at_nat @ Z @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3028_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ Z @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3029_sup__least,axiom,
    ! [Y: filter_nat,X: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( ord_le2510731241096832064er_nat @ Z @ X )
       => ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3030_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3031_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3032_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_3033_le__iff__sup,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3034_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y5: assn] :
          ( ( sup_sup_assn @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3035_le__iff__sup,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [X2: filter_nat,Y5: filter_nat] :
          ( ( sup_sup_filter_nat @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3036_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y5: set_nat] :
          ( ( sup_sup_set_nat @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3037_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( sup_sup_nat @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3038_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y5: int] :
          ( ( sup_sup_int @ X2 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_3039_sup_OorderE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3040_sup_OorderE,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( A2
        = ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3041_sup_OorderE,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3042_sup_OorderE,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3043_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3044_sup_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( A2
        = ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_3045_sup_OorderI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
     => ( ord_le1268244103169919719at_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3046_sup_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2
        = ( sup_sup_assn @ A2 @ B2 ) )
     => ( ord_less_eq_assn @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3047_sup_OorderI,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2
        = ( sup_sup_filter_nat @ A2 @ B2 ) )
     => ( ord_le2510731241096832064er_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3048_sup_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3049_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3050_sup_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( sup_sup_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_3051_sup__unique,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ Y2 @ X3 )
             => ( ( ord_le1268244103169919719at_nat @ Z4 @ X3 )
               => ( ord_le1268244103169919719at_nat @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_su5525570899277871387at_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3052_sup__unique,axiom,
    ! [F4: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y2: assn] : ( ord_less_eq_assn @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: assn,Y2: assn] : ( ord_less_eq_assn @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: assn,Y2: assn,Z4: assn] :
              ( ( ord_less_eq_assn @ Y2 @ X3 )
             => ( ( ord_less_eq_assn @ Z4 @ X3 )
               => ( ord_less_eq_assn @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_assn @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3053_sup__unique,axiom,
    ! [F4: filter_nat > filter_nat > filter_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X3: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: filter_nat,Y2: filter_nat,Z4: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ Y2 @ X3 )
             => ( ( ord_le2510731241096832064er_nat @ Z4 @ X3 )
               => ( ord_le2510731241096832064er_nat @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_filter_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3054_sup__unique,axiom,
    ! [F4: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: set_nat,Y2: set_nat,Z4: set_nat] :
              ( ( ord_less_eq_set_nat @ Y2 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z4 @ X3 )
               => ( ord_less_eq_set_nat @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3055_sup__unique,axiom,
    ! [F4: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: nat,Y2: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ Y2 @ X3 )
             => ( ( ord_less_eq_nat @ Z4 @ X3 )
               => ( ord_less_eq_nat @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3056_sup__unique,axiom,
    ! [F4: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ X3 @ ( F4 @ X3 @ Y2 ) )
     => ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F4 @ X3 @ Y2 ) )
       => ( ! [X3: int,Y2: int,Z4: int] :
              ( ( ord_less_eq_int @ Y2 @ X3 )
             => ( ( ord_less_eq_int @ Z4 @ X3 )
               => ( ord_less_eq_int @ ( F4 @ Y2 @ Z4 ) @ X3 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F4 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_3057_sup_Oabsorb1,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3058_sup_Oabsorb1,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3059_sup_Oabsorb1,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( sup_sup_filter_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3060_sup_Oabsorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3061_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3062_sup_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_3063_sup_Oabsorb2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3064_sup_Oabsorb2,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3065_sup_Oabsorb2,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A2 @ B2 )
     => ( ( sup_sup_filter_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3066_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3067_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3068_sup_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_3069_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X )
     => ( ( sup_su5525570899277871387at_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3070_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( sup_sup_assn @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3071_sup__absorb1,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( sup_sup_filter_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3072_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( sup_sup_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3073_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3074_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_3075_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ Y )
     => ( ( sup_su5525570899277871387at_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3076_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( sup_sup_assn @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3077_sup__absorb2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( sup_sup_filter_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3078_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( sup_sup_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3079_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3080_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_3081_sup_OboundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
         => ~ ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3082_sup_OboundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_assn @ B2 @ A2 )
         => ~ ( ord_less_eq_assn @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3083_sup_OboundedE,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
         => ~ ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3084_sup_OboundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3085_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3086_sup_OboundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B2 @ A2 )
         => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_3087_sup_OboundedI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ C @ A2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3088_sup_OboundedI,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C @ A2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3089_sup_OboundedI,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B2 @ A2 )
     => ( ( ord_le2510731241096832064er_nat @ C @ A2 )
       => ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3090_sup_OboundedI,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ A2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3091_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3092_sup_OboundedI,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ A2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_3093_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( A3
          = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3094_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( A3
          = ( sup_sup_assn @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3095_sup_Oorder__iff,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( A3
          = ( sup_sup_filter_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3096_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( A3
          = ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3097_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( A3
          = ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3098_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( A3
          = ( sup_sup_int @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_3099_sup_Ocobounded1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3100_sup_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3101_sup_Ocobounded1,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3102_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3103_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3104_sup_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_3105_sup_Ocobounded2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3106_sup_Ocobounded2,axiom,
    ! [B2: assn,A2: assn] : ( ord_less_eq_assn @ B2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3107_sup_Ocobounded2,axiom,
    ! [B2: filter_nat,A2: filter_nat] : ( ord_le2510731241096832064er_nat @ B2 @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3108_sup_Ocobounded2,axiom,
    ! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3109_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3110_sup_Ocobounded2,axiom,
    ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_3111_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3112_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( sup_sup_assn @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3113_sup_Oabsorb__iff1,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( sup_sup_filter_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3114_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3115_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( sup_sup_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3116_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( sup_sup_int @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_3117_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3118_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( sup_sup_assn @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3119_sup_Oabsorb__iff2,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( sup_sup_filter_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3120_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3121_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( sup_sup_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3122_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( sup_sup_int @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_3123_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A2 )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3124_sup_OcoboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C @ A2 )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3125_sup_OcoboundedI1,axiom,
    ! [C: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ C @ A2 )
     => ( ord_le2510731241096832064er_nat @ C @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3126_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3127_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3128_sup_OcoboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_3129_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ B2 )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3130_sup_OcoboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ C @ B2 )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3131_sup_OcoboundedI2,axiom,
    ! [C: filter_nat,B2: filter_nat,A2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ C @ B2 )
     => ( ord_le2510731241096832064er_nat @ C @ ( sup_sup_filter_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3132_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B2 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3133_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3134_sup_OcoboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ C @ B2 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_3135_inf__min,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_min
thf(fact_3136_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ C @ B2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_3137_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ B2 )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_3138_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ C @ B2 )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_3139_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_3140_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_int @ C @ B2 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_3141_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ C @ A2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_3142_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ A2 )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_3143_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ C @ A2 )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_3144_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_3145_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ A2 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_3146_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( A3
            = ( sup_sup_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_3147_sup_Ostrict__order__iff,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( ( A3
            = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_3148_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( A3
            = ( sup_sup_assn @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_3149_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( sup_sup_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_3150_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( sup_sup_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_3151_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_nat @ B2 @ A2 )
         => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_3152_sup_Ostrict__boundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le2604355607129572851at_nat @ B2 @ A2 )
         => ~ ( ord_le2604355607129572851at_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_3153_sup_Ostrict__boundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_assn @ B2 @ A2 )
         => ~ ( ord_less_assn @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_3154_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_3155_sup_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_int @ B2 @ A2 )
         => ~ ( ord_less_int @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_3156_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_3157_sup_Oabsorb4,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A2 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_3158_sup_Oabsorb4,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_3159_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_3160_sup_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_3161_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_3162_sup_Oabsorb3,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ B2 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_3163_sup_Oabsorb3,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_3164_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_3165_sup_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_3166_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ X @ B2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_3167_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ B2 )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_3168_less__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ X @ B2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_3169_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_3170_less__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_int @ X @ B2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_3171_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ X @ A2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_3172_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ A2 )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_3173_less__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ X @ A2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_3174_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_3175_less__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_3176_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3177_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3178_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3179_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3180_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3181_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_3182_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ Y2 @ Z4 ) )
          = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y2 ) @ ( inf_in2572325071724192079at_nat @ X3 @ Z4 ) ) )
     => ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
        = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_3183_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y2: set_nat,Z4: set_nat] :
          ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z4 ) )
          = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z4 ) ) )
     => ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_3184_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y2: assn,Z4: assn] :
          ( ( inf_inf_assn @ X3 @ ( sup_sup_assn @ Y2 @ Z4 ) )
          = ( sup_sup_assn @ ( inf_inf_assn @ X3 @ Y2 ) @ ( inf_inf_assn @ X3 @ Z4 ) ) )
     => ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
        = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_3185_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
          ( ( inf_in7913087082777306421at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ Y2 @ Z4 ) )
          = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ Y2 ) @ ( inf_in7913087082777306421at_nat @ X3 @ Z4 ) ) )
     => ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
        = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_3186_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
          ( ( sup_su6327502436637775413at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y2 @ Z4 ) )
          = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X3 @ Y2 ) @ ( sup_su6327502436637775413at_nat @ X3 @ Z4 ) ) )
     => ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
        = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_3187_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y2: set_nat,Z4: set_nat] :
          ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z4 ) )
          = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z4 ) ) )
     => ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
        = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_3188_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y2: assn,Z4: assn] :
          ( ( sup_sup_assn @ X3 @ ( inf_inf_assn @ Y2 @ Z4 ) )
          = ( inf_inf_assn @ ( sup_sup_assn @ X3 @ Y2 ) @ ( sup_sup_assn @ X3 @ Z4 ) ) )
     => ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
        = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_3189_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ Y2 @ Z4 ) )
          = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y2 ) @ ( sup_su5525570899277871387at_nat @ X3 @ Z4 ) ) )
     => ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
        = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_3190_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_3191_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_3192_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_3193_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_3194_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3195_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3196_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3197_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3198_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_3199_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_3200_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_3201_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_3202_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3203_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3204_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3205_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z ) @ X )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3206_Un__empty__left,axiom,
    ! [B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_3207_Un__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_3208_Un__empty__left,axiom,
    ! [B: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B )
      = B ) ).

% Un_empty_left
thf(fact_3209_Un__empty__left,axiom,
    ! [B: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_3210_Un__empty__left,axiom,
    ! [B: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B )
      = B ) ).

% Un_empty_left
thf(fact_3211_Un__empty__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% Un_empty_right
thf(fact_3212_Un__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% Un_empty_right
thf(fact_3213_Un__empty__right,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Un_empty_right
thf(fact_3214_Un__empty__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_3215_Un__empty__right,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Un_empty_right
thf(fact_3216_foldl__un__empty__eq,axiom,
    ! [I: set_Pr4329608150637261639at_nat,Ww: list_s4242580248156217677at_nat] :
      ( ( foldl_4541198706405875041at_nat @ sup_su5525570899277871387at_nat @ I @ Ww )
      = ( sup_su5525570899277871387at_nat @ I @ ( foldl_4541198706405875041at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_3217_foldl__un__empty__eq,axiom,
    ! [I: set_Pr1261947904930325089at_nat,Ww: list_s1210847774152347623at_nat] :
      ( ( foldl_5307762784932486881at_nat @ sup_su6327502436637775413at_nat @ I @ Ww )
      = ( sup_su6327502436637775413at_nat @ I @ ( foldl_5307762784932486881at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ Ww ) ) ) ).

% foldl_un_empty_eq
thf(fact_3218_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_3219_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_3220_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_3221_Un__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B @ D )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_3222_Un__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_3223_Un__least,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B @ C2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_3224_Un__least,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_3225_Un__upper1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_3226_Un__upper1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_3227_Un__upper2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_3228_Un__upper2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_3229_Un__absorb1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_3230_Un__absorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_3231_Un__absorb2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_3232_Un__absorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_3233_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A @ B ) )
     => ~ ! [A8: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A8 @ A )
           => ! [B8: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B8 @ B )
               => ( C2
                 != ( sup_su5525570899277871387at_nat @ A8 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_3234_subset__UnE,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
     => ~ ! [A8: set_nat] :
            ( ( ord_less_eq_set_nat @ A8 @ A )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_nat @ A8 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_3235_subset__Un__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_3236_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_3237_Un__Int__assoc__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ A @ ( sup_su6327502436637775413at_nat @ B @ C2 ) ) )
      = ( ord_le3146513528884898305at_nat @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_3238_Un__Int__assoc__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A @ B ) @ C2 )
        = ( inf_in7913087082777306421at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) )
      = ( ord_le1268244103169919719at_nat @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_3239_Un__Int__assoc__eq,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
        = ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
      = ( ord_less_eq_set_nat @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_3240_set__diff__diff__left,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ C2 )
      = ( minus_3314409938677909166at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_3241_set__diff__diff__left,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
      = ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_3242_Un__Diff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A @ C2 ) @ ( minus_3314409938677909166at_nat @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_3243_Un__Diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_3244_Diff__Un,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( sup_su6327502436637775413at_nat @ B @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ A @ C2 ) ) ) ).

% Diff_Un
thf(fact_3245_Diff__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ A @ C2 ) ) ) ).

% Diff_Un
thf(fact_3246_Diff__Un,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).

% Diff_Un
thf(fact_3247_Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ A @ C2 ) ) ) ).

% Diff_Int
thf(fact_3248_Diff__Int,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( inf_in7913087082777306421at_nat @ B @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ A @ C2 ) ) ) ).

% Diff_Int
thf(fact_3249_Diff__Int,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).

% Diff_Int
thf(fact_3250_Int__Diff__Un,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = A ) ).

% Int_Diff_Un
thf(fact_3251_Int__Diff__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A @ B ) @ ( minus_3314409938677909166at_nat @ A @ B ) )
      = A ) ).

% Int_Diff_Un
thf(fact_3252_Int__Diff__Un,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
      = A ) ).

% Int_Diff_Un
thf(fact_3253_Un__Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = A ) ).

% Un_Diff_Int
thf(fact_3254_Un__Diff__Int,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ ( inf_in7913087082777306421at_nat @ A @ B ) )
      = A ) ).

% Un_Diff_Int
thf(fact_3255_Un__Diff__Int,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
      = A ) ).

% Un_Diff_Int
thf(fact_3256_in__set__butlastD,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ ( butlas2561766218336428473at_nat @ Xs ) ) )
     => ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3257_in__set__butlastD,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( butlas5569151904373679443at_nat @ Xs ) ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3258_in__set__butlastD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ ( butlas8287983287669047583_nat_o @ Xs ) ) )
     => ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3259_in__set__butlastD,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ ( butlast_int @ Xs ) ) )
     => ( member_int @ X @ ( set_int2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3260_in__set__butlastD,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3261_in__set__butlastD,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( butlas3012047794866324995n_assn @ Xs ) ) )
     => ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_3262_distinct__butlast,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ ( butlast_int @ Xs ) ) ) ).

% distinct_butlast
thf(fact_3263_distinct__butlast,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( butlast_nat @ Xs ) ) ) ).

% distinct_butlast
thf(fact_3264_butlast__update_H,axiom,
    ! [L: list_nat,I: nat,X: nat] :
      ( ( list_update_nat @ ( butlast_nat @ L ) @ I @ X )
      = ( butlast_nat @ ( list_update_nat @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_3265_butlast__update_H,axiom,
    ! [L: list_c,I: nat,X: c] :
      ( ( list_update_c @ ( butlast_c @ L ) @ I @ X )
      = ( butlast_c @ ( list_update_c @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_3266_butlast__update_H,axiom,
    ! [L: list_d,I: nat,X: d] :
      ( ( list_update_d @ ( butlast_d @ L ) @ I @ X )
      = ( butlast_d @ ( list_update_d @ L @ I @ X ) ) ) ).

% butlast_update'
thf(fact_3267_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).

% sorted_rev_iff_nth_mono
thf(fact_3268_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ J2 ) @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).

% sorted_rev_iff_nth_mono
thf(fact_3269_sorted__rev__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).

% sorted_rev_nth_mono
thf(fact_3270_sorted__rev__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ J ) @ ( nth_int @ Xs @ I ) ) ) ) ) ).

% sorted_rev_nth_mono
thf(fact_3271_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_Code_integer] :
      ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ Xs )
     => ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3272_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3273_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3274_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3275_strict__sorted__equal,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer] :
      ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ Xs )
     => ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ Ys )
       => ( ( ( set_Code_integer2 @ Ys )
            = ( set_Code_integer2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_3276_strict__sorted__equal,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ Xs )
     => ( ( sorted_wrt_num @ ord_less_num @ Ys )
       => ( ( ( set_num2 @ Ys )
            = ( set_num2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_3277_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
       => ( ( ( set_nat2 @ Ys )
            = ( set_nat2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_3278_strict__sorted__equal,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( ( sorted_wrt_int @ ord_less_int @ Ys )
       => ( ( ( set_int2 @ Ys )
            = ( set_int2 @ Xs ) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_3279_sorted__wrt01,axiom,
    ! [Xs: list_c,P: c > c > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_c @ Xs ) @ one_one_nat )
     => ( sorted_wrt_c @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3280_sorted__wrt01,axiom,
    ! [Xs: list_d,P: d > d > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_d @ Xs ) @ one_one_nat )
     => ( sorted_wrt_d @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3281_sorted__wrt01,axiom,
    ! [Xs: list_nat,P: nat > nat > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3282_sorted__wrt01,axiom,
    ! [Xs: list_int,P: int > int > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3283_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3284_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3285_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) @ ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3286_distrib__inf__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ ( inf_inf_filter_nat @ X @ Y ) @ ( inf_inf_filter_nat @ X @ Z ) ) @ ( inf_inf_filter_nat @ X @ ( sup_sup_filter_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3287_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3288_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3289_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_3290_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3291_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3292_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) @ ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3293_distrib__sup__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) ) @ ( inf_inf_filter_nat @ ( sup_sup_filter_nat @ X @ Y ) @ ( sup_sup_filter_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3294_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3295_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3296_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_3297_sorted__remove1,axiom,
    ! [Xs: list_nat,A2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A2 @ Xs ) ) ) ).

% sorted_remove1
thf(fact_3298_sorted__remove1,axiom,
    ! [Xs: list_int,A2: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( remove1_int @ A2 @ Xs ) ) ) ).

% sorted_remove1
thf(fact_3299_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_3300_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_3301_insert__is__Un,axiom,
    ( insert_o2
    = ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o2 @ A3 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_3302_insert__is__Un,axiom,
    ( insert_nat2
    = ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A3 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_3303_insert__is__Un,axiom,
    ( insert_int2
    = ( ^ [A3: int] : ( sup_sup_set_int @ ( insert_int2 @ A3 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_3304_Un__singleton__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B )
        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_3305_Un__singleton__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B )
        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_3306_Un__singleton__iff,axiom,
    ! [A: set_o,B: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = ( insert_o2 @ X @ bot_bot_set_o ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o2 @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o2 @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o2 @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_3307_Un__singleton__iff,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_3308_Un__singleton__iff,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = ( insert_int2 @ X @ bot_bot_set_int ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int2 @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int2 @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int2 @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_3309_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_3310_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A @ B ) )
      = ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_3311_singleton__Un__iff,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ( insert_o2 @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o2 @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o2 @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o2 @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_3312_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ( insert_nat2 @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_3313_singleton__Un__iff,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( ( insert_int2 @ X @ bot_bot_set_int )
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int2 @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int2 @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int2 @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_3314_Diff__partition,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( sup_su5525570899277871387at_nat @ A @ ( minus_3314409938677909166at_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_3315_Diff__partition,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_3316_Diff__subset__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ C2 )
      = ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_3317_Diff__subset__conv,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
      = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_3318_strict__sorted__iff,axiom,
    ! [L: list_Code_integer] :
      ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ L )
      = ( ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ L )
        & ( distin1543349897113766820nteger @ L ) ) ) ).

% strict_sorted_iff
thf(fact_3319_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_3320_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_3321_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_3322_sorted01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% sorted01
thf(fact_3323_sorted01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% sorted01
thf(fact_3324_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
         => ( ( distinct_nat @ Ys )
           => ( ( ( set_nat2 @ Xs )
                = ( set_nat2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_3325_sorted__distinct__set__unique,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( distinct_int @ Xs )
       => ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
         => ( ( distinct_int @ Ys )
           => ( ( ( set_int2 @ Xs )
                = ( set_int2 @ Ys ) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_3326_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_c
    = ( ^ [P2: c > c > $o,Xs3: list_c] :
        ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_c @ Xs3 ) )
           => ( P2 @ ( nth_c @ Xs3 @ I4 ) @ ( nth_c @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3327_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_d
    = ( ^ [P2: d > d > $o,Xs3: list_d] :
        ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_d @ Xs3 ) )
           => ( P2 @ ( nth_d @ Xs3 @ I4 ) @ ( nth_d @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3328_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P2: nat > nat > $o,Xs3: list_nat] :
        ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
           => ( P2 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3329_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P2: int > int > $o,Xs3: list_int] :
        ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs3 ) )
           => ( P2 @ ( nth_int @ Xs3 @ I4 ) @ ( nth_int @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3330_sorted__wrt__nth__less,axiom,
    ! [P: c > c > $o,Xs: list_c,I: nat,J: nat] :
      ( ( sorted_wrt_c @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_c @ Xs ) )
         => ( P @ ( nth_c @ Xs @ I ) @ ( nth_c @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3331_sorted__wrt__nth__less,axiom,
    ! [P: d > d > $o,Xs: list_d,I: nat,J: nat] :
      ( ( sorted_wrt_d @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_d @ Xs ) )
         => ( P @ ( nth_d @ Xs @ I ) @ ( nth_d @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3332_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3333_sorted__wrt__nth__less,axiom,
    ! [P: int > int > $o,Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3334_nth__butlast,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( ord_less_nat @ N @ ( size_size_list_c @ ( butlast_c @ Xs ) ) )
     => ( ( nth_c @ ( butlast_c @ Xs ) @ N )
        = ( nth_c @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_3335_nth__butlast,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( ord_less_nat @ N @ ( size_size_list_d @ ( butlast_d @ Xs ) ) )
     => ( ( nth_d @ ( butlast_d @ Xs ) @ N )
        = ( nth_d @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_3336_nth__butlast,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
     => ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_3337_nth__butlast,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ ( butlast_int @ Xs ) ) )
     => ( ( nth_int @ ( butlast_int @ Xs ) @ N )
        = ( nth_int @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_3338_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3339_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3340_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3341_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3342_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3343_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J2: nat] :
            ( ( ord_less_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3344_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_3345_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_3346_times__assn__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( times_assn_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ? [As1: set_nat,As22: set_nat] :
            ( ( As2
              = ( sup_sup_set_nat @ As1 @ As22 ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( P @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
            & ( Q @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_3347_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As22: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As1 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_3348_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ~ ? [As12: set_nat,As23: set_nat] :
                  ( ( As
                    = ( sup_sup_set_nat @ As12 @ As23 ) )
                  & ( ( inf_inf_set_nat @ As12 @ As23 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_3349_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ? [As13: set_nat,As24: set_nat] :
                ( ( As
                  = ( sup_sup_set_nat @ As13 @ As24 ) )
                & ( ( inf_inf_set_nat @ As13 @ As24 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_3350_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I6: nat] :
                  ( ( ord_less_nat @ K @ I6 )
                 => ( P @ I6 ) )
             => ( P @ K ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_3351_merge__correct,axiom,
    ! [L12: list_nat,L23: list_nat] :
      ( ( ( distinct_nat @ L12 )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ L12 ) )
     => ( ( ( distinct_nat @ L23 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ L23 ) )
       => ( ( distinct_nat @ ( merge_nat @ L12 @ L23 ) )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ ( merge_nat @ L12 @ L23 ) )
          & ( ( set_nat2 @ ( merge_nat @ L12 @ L23 ) )
            = ( sup_sup_set_nat @ ( set_nat2 @ L12 ) @ ( set_nat2 @ L23 ) ) ) ) ) ) ).

% merge_correct
thf(fact_3352_merge__correct,axiom,
    ! [L12: list_int,L23: list_int] :
      ( ( ( distinct_int @ L12 )
        & ( sorted_wrt_int @ ord_less_eq_int @ L12 ) )
     => ( ( ( distinct_int @ L23 )
          & ( sorted_wrt_int @ ord_less_eq_int @ L23 ) )
       => ( ( distinct_int @ ( merge_int @ L12 @ L23 ) )
          & ( sorted_wrt_int @ ord_less_eq_int @ ( merge_int @ L12 @ L23 ) )
          & ( ( set_int2 @ ( merge_int @ L12 @ L23 ) )
            = ( sup_sup_set_int @ ( set_int2 @ L12 ) @ ( set_int2 @ L23 ) ) ) ) ) ) ).

% merge_correct
thf(fact_3353_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_c,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_c @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_c @ ( slice_c @ From @ To @ Xs ) @ I )
            = ( nth_c @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3354_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_d,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_d @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_d @ ( slice_d @ From @ To @ Xs ) @ I )
            = ( nth_d @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3355_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_nat @ ( slice_nat @ From @ To @ Xs ) @ I )
            = ( nth_nat @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3356_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_int @ ( slice_int @ From @ To @ Xs ) @ I )
            = ( nth_int @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3357_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I4 ) ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_3358_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I4 ) ) @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_3359_add__right__cancel,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = ( plus_p7104986032573967614at_nat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_3360_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_3361_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_3362_add__right__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_3363_add__left__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = ( plus_p7104986032573967614at_nat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_3364_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_3365_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_3366_add__left__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_3367_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        | ( rep_assn @ Q @ H2 ) ) ) ).

% mod_or_dist
thf(fact_3368_merge__pure__or,axiom,
    ! [A2: $o,B2: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          | B2 ) ) ) ).

% merge_pure_or
thf(fact_3369_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_3370_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_3371_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_3372_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_3373_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_3374_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_3375_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_3376_add__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_3377_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_3378_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_3379_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_3380_add__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_3381_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_3382_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_3383_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3384_add__diff__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3385_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3386_diff__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3387_add__diff__cancel__left,axiom,
    ! [C: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ C @ A2 ) @ ( plus_p7104986032573967614at_nat @ C @ B2 ) )
      = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3388_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3389_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3390_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3391_add__diff__cancel__left_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3392_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3393_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3394_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_3395_add__diff__cancel__right,axiom,
    ! [A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ C ) @ ( plus_p7104986032573967614at_nat @ B2 @ C ) )
      = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3396_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3397_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3398_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3399_add__diff__cancel__right_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3400_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3401_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3402_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3403_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_3404_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_3405_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_3406_nat__add__left__cancel__less,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_3407_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3408_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3409_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3410_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3411_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3412_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3413_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ ( suc @ A2 ) @ B2 )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(1)
thf(fact_3414_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ B2 @ ( suc @ A2 ) )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(2)
thf(fact_3415_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        | ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_3416_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_3417_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q5: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q5 ) ) ) ) ).

% less_natE
thf(fact_3418_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_3419_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_3420_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N2: nat] :
        ? [K4: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_3421_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_3422_add__right__imp__eq,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = ( plus_p7104986032573967614at_nat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_3423_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_3424_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_3425_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_3426_add__left__imp__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = ( plus_p7104986032573967614at_nat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_3427_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_3428_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_3429_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_3430_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ B2 @ ( plus_p7104986032573967614at_nat @ A2 @ C ) )
      = ( plus_p7104986032573967614at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3431_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3432_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3433_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3434_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p7104986032573967614at_nat
    = ( ^ [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] : ( plus_p7104986032573967614at_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3435_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3436_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3437_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3438_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_3439_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_3440_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_3441_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_3442_add_Oassoc,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ C )
      = ( plus_p7104986032573967614at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_3443_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_3444_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_3445_add_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_3446_group__cancel_Oadd2,axiom,
    ! [B: multis2468970476368604999at_nat,K3: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( B
        = ( plus_p7104986032573967614at_nat @ K3 @ B2 ) )
     => ( ( plus_p7104986032573967614at_nat @ A2 @ B )
        = ( plus_p7104986032573967614at_nat @ K3 @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_3447_group__cancel_Oadd2,axiom,
    ! [B: rat,K3: rat,B2: rat,A2: rat] :
      ( ( B
        = ( plus_plus_rat @ K3 @ B2 ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_3448_group__cancel_Oadd2,axiom,
    ! [B: nat,K3: nat,B2: nat,A2: nat] :
      ( ( B
        = ( plus_plus_nat @ K3 @ B2 ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_3449_group__cancel_Oadd2,axiom,
    ! [B: int,K3: int,B2: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ K3 @ B2 ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_3450_group__cancel_Oadd1,axiom,
    ! [A: multis2468970476368604999at_nat,K3: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ K3 @ A2 ) )
     => ( ( plus_p7104986032573967614at_nat @ A @ B2 )
        = ( plus_p7104986032573967614at_nat @ K3 @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_3451_group__cancel_Oadd1,axiom,
    ! [A: rat,K3: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K3 @ A2 ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_3452_group__cancel_Oadd1,axiom,
    ! [A: nat,K3: nat,A2: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ K3 @ A2 ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_3453_group__cancel_Oadd1,axiom,
    ! [A: int,K3: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K3 @ A2 ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_3454_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( ( plus_plus_rat @ I @ K3 )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3455_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( ( plus_plus_nat @ I @ K3 )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3456_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( ( plus_plus_int @ I @ K3 )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3457_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_3458_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_3459_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_3460_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_3461_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_3462_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_3463_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C5: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_3464_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_3465_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_3466_add__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_3467_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ~ ! [C4: nat] :
            ( B2
           != ( plus_plus_nat @ A2 @ C4 ) ) ) ).

% less_eqE
thf(fact_3468_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_3469_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_3470_add__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_3471_add__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_3472_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_3473_add__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_3474_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K3 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3475_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K3 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3476_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K3 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3477_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K3 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3478_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K3 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3479_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K3 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3480_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3481_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3482_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3483_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K3 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3484_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K3 @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3485_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K3 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3486_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K3 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3487_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K3 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3488_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( I = J )
        & ( ord_le6747313008572928689nteger @ K3 @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3489_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K3 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3490_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K3 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3491_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3492_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( K3 = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3493_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3494_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K3 = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3495_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3496_add__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3497_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3498_add__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3499_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_3500_add__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_3501_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_3502_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_3503_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3504_add__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3505_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3506_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3507_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_3508_add__less__imp__less__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B2 ) )
     => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_3509_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_3510_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_3511_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_3512_add__less__imp__less__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
     => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_3513_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_3514_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_3515_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3516_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3517_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3518_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3519_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3520_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3521_comm__semiring__class_Odistrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3522_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3523_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3524_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3525_distrib__left,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_3526_distrib__left,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_3527_distrib__left,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_3528_distrib__left,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_3529_distrib__right,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_3530_distrib__right,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_3531_distrib__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_3532_distrib__right,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_3533_combine__common__factor,axiom,
    ! [A2: code_integer,E: code_integer,B2: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3534_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3535_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3536_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_3537_group__cancel_Osub1,axiom,
    ! [A: rat,K3: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K3 @ A2 ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ K3 @ ( minus_minus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3538_group__cancel_Osub1,axiom,
    ! [A: int,K3: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K3 @ A2 ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ K3 @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3539_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = C )
      = ( A2
        = ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3540_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = C )
      = ( A2
        = ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3541_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( A2
        = ( minus_minus_rat @ C @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3542_eq__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( A2
        = ( minus_minus_int @ C @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3543_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_3544_add__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_3545_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3546_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3547_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3548_diff__add__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3549_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3550_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3551_add__implies__diff,axiom,
    ! [C: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3552_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3553_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3554_add__implies__diff,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ( plus_plus_int @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_int @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3555_diff__diff__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ A2 @ B2 ) @ C )
      = ( minus_4286766774447292334at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3556_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3557_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3558_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3559_star__or__dist1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A @ B ) @ C2 )
      = ( sup_sup_assn @ ( times_times_assn @ A @ C2 ) @ ( times_times_assn @ B @ C2 ) ) ) ).

% star_or_dist1
thf(fact_3560_star__or__dist2,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( ( times_times_assn @ C2 @ ( sup_sup_assn @ A @ B ) )
      = ( sup_sup_assn @ ( times_times_assn @ C2 @ A ) @ ( times_times_assn @ C2 @ B ) ) ) ).

% star_or_dist2
thf(fact_3561_Nat_OlessE,axiom,
    ! [I: nat,K3: nat] :
      ( ( ord_less_nat @ I @ K3 )
     => ( ( K3
         != ( suc @ I ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( K3
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_3562_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_3563_Suc__lessE,axiom,
    ! [I: nat,K3: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K3 )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I @ J3 )
           => ( K3
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_3564_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_3565_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_3566_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_3567_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_3568_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_3569_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_3570_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_3571_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M4: nat] :
            ( ( M2
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_3572_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_3573_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_3574_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K3 )
       => ( ord_less_nat @ ( suc @ I ) @ K3 ) ) ) ).

% less_trans_Suc
thf(fact_3575_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
       => ( ! [I5: nat,J3: nat,K: nat] :
              ( ( ord_less_nat @ I5 @ J3 )
             => ( ( ord_less_nat @ J3 @ K )
               => ( ( P @ I5 @ J3 )
                 => ( ( P @ J3 @ K )
                   => ( P @ I5 @ K ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_3576_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I5: nat] :
            ( ( J
              = ( suc @ I5 ) )
           => ( P @ I5 ) )
       => ( ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ J )
             => ( ( P @ ( suc @ I5 ) )
               => ( P @ I5 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_3577_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_3578_add__lessD1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
     => ( ord_less_nat @ I @ K3 ) ) ).

% add_lessD1
thf(fact_3579_add__less__mono,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K3 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_3580_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_3581_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_3582_add__less__mono1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).

% add_less_mono1
thf(fact_3583_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_3584_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_3585_less__add__eq__less,axiom,
    ! [K3: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K3 @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K3 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_3586_ent__disjI2__direct,axiom,
    ! [B: assn,A: assn] : ( entails @ B @ ( sup_sup_assn @ A @ B ) ) ).

% ent_disjI2_direct
thf(fact_3587_ent__disjI1__direct,axiom,
    ! [A: assn,B: assn] : ( entails @ A @ ( sup_sup_assn @ A @ B ) ) ).

% ent_disjI1_direct
thf(fact_3588_ent__disjI2_H,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( entails @ A @ C2 )
     => ( entails @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% ent_disjI2'
thf(fact_3589_ent__disjI1_H,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entails @ A @ B )
     => ( entails @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% ent_disjI1'
thf(fact_3590_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ Q @ R ) ) ).

% ent_disjI2
thf(fact_3591_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ P @ R ) ) ).

% ent_disjI1
thf(fact_3592_ent__disjE,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( entails @ A @ C2 )
     => ( ( entails @ B @ C2 )
       => ( entails @ ( sup_sup_assn @ A @ B ) @ C2 ) ) ) ).

% ent_disjE
thf(fact_3593_min__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_min_rat @ Y @ Z ) )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_3594_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_3595_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_3596_min__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_3597_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_3598_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_3599_entt__disjE,axiom,
    ! [A: assn,M5: assn,B: assn] :
      ( ( entailst @ A @ M5 )
     => ( ( entailst @ B @ M5 )
       => ( entailst @ ( sup_sup_assn @ A @ B ) @ M5 ) ) ) ).

% entt_disjE
thf(fact_3600_entt__disjD1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entailst @ ( sup_sup_assn @ A @ B ) @ C2 )
     => ( entailst @ A @ C2 ) ) ).

% entt_disjD1
thf(fact_3601_entt__disjD2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entailst @ ( sup_sup_assn @ A @ B ) @ C2 )
     => ( entailst @ B @ C2 ) ) ).

% entt_disjD2
thf(fact_3602_entt__disjI1_H,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( entailst @ A @ B )
     => ( entailst @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% entt_disjI1'
thf(fact_3603_entt__disjI2_H,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( entailst @ A @ C2 )
     => ( entailst @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% entt_disjI2'
thf(fact_3604_entt__disjI1__direct,axiom,
    ! [A: assn,B: assn] : ( entailst @ A @ ( sup_sup_assn @ A @ B ) ) ).

% entt_disjI1_direct
thf(fact_3605_entt__disjI2__direct,axiom,
    ! [B: assn,A: assn] : ( entailst @ B @ ( sup_sup_assn @ A @ B ) ) ).

% entt_disjI2_direct
thf(fact_3606_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% norm_assertion_simps(6)
thf(fact_3607_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% norm_assertion_simps(5)
thf(fact_3608_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K3 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3609_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K3 @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3610_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K3 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3611_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K3 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3612_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K3 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3613_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le3102999989581377725nteger @ K3 @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3614_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K3 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3615_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K3 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3616_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3617_add__le__less__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3618_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3619_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3620_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3621_add__less__le__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3622_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3623_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3624_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3625_diff__le__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3626_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_3627_le__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_3628_add__le__imp__le__diff,axiom,
    ! [I: rat,K3: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K3 ) ) ) ).

% add_le_imp_le_diff
thf(fact_3629_add__le__imp__le__diff,axiom,
    ! [I: nat,K3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K3 ) ) ) ).

% add_le_imp_le_diff
thf(fact_3630_add__le__imp__le__diff,axiom,
    ! [I: int,K3: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K3 ) ) ) ).

% add_le_imp_le_diff
thf(fact_3631_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_3632_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K3: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K3 ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K3 ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K3 ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K3 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3633_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K3: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K3 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K3 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K3 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3634_add__le__add__imp__diff__le,axiom,
    ! [I: int,K3: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K3 ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K3 ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K3 ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K3 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3635_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_3636_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3637_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3638_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3639_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3640_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3641_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3642_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3643_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A2 )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3644_less__add__one,axiom,
    ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).

% less_add_one
thf(fact_3645_less__add__one,axiom,
    ! [A2: code_integer] : ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_3646_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_3647_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_3648_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3649_add__mono1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B2 @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_3650_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3651_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3652_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3653_diff__less__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C )
      = ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3654_diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3655_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_3656_less__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_3657_less__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_3658_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ ( ord_less_rat @ A2 @ B2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3659_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( plus_p5714425477246183910nteger @ B2 @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3660_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3661_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ ( ord_less_int @ A2 @ B2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3662_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3663_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3664_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3665_eq__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( C
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3666_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3667_eq__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3668_eq__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3669_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3670_eq__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3671_lift__Suc__mono__less,axiom,
    ! [F4: nat > code_integer,N: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_le6747313008572928689nteger @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ord_le6747313008572928689nteger @ ( F4 @ N ) @ ( F4 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_3672_lift__Suc__mono__less,axiom,
    ! [F4: nat > assn,N: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_assn @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ord_less_assn @ ( F4 @ N ) @ ( F4 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_3673_lift__Suc__mono__less,axiom,
    ! [F4: nat > num,N: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_num @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ord_less_num @ ( F4 @ N ) @ ( F4 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_3674_lift__Suc__mono__less,axiom,
    ! [F4: nat > nat,N: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_nat @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ord_less_nat @ ( F4 @ N ) @ ( F4 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_3675_lift__Suc__mono__less,axiom,
    ! [F4: nat > int,N: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_int @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ord_less_int @ ( F4 @ N ) @ ( F4 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_3676_lift__Suc__mono__less__iff,axiom,
    ! [F4: nat > code_integer,N: nat,M2: nat] :
      ( ! [N5: nat] : ( ord_le6747313008572928689nteger @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_le6747313008572928689nteger @ ( F4 @ N ) @ ( F4 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_3677_lift__Suc__mono__less__iff,axiom,
    ! [F4: nat > assn,N: nat,M2: nat] :
      ( ! [N5: nat] : ( ord_less_assn @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_assn @ ( F4 @ N ) @ ( F4 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_3678_lift__Suc__mono__less__iff,axiom,
    ! [F4: nat > num,N: nat,M2: nat] :
      ( ! [N5: nat] : ( ord_less_num @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_num @ ( F4 @ N ) @ ( F4 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_3679_lift__Suc__mono__less__iff,axiom,
    ! [F4: nat > nat,N: nat,M2: nat] :
      ( ! [N5: nat] : ( ord_less_nat @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ ( F4 @ N ) @ ( F4 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_3680_lift__Suc__mono__less__iff,axiom,
    ! [F4: nat > int,N: nat,M2: nat] :
      ( ! [N5: nat] : ( ord_less_int @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) )
     => ( ( ord_less_int @ ( F4 @ N ) @ ( F4 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_3681_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_3682_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_3683_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I @ N5 )
             => ( ( ord_less_nat @ N5 @ J )
               => ( ( P @ N5 )
                 => ( P @ ( suc @ N5 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_3684_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I @ N5 )
             => ( ( ord_less_nat @ N5 @ J )
               => ( ( P @ ( suc @ N5 ) )
                 => ( P @ N5 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_3685_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_3686_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_3687_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_3688_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_3689_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_3690_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2
        = ( suc @ A2 ) ) ) ).

% nat_in_between_eq(1)
thf(fact_3691_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
thf(fact_3692_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_3693_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_3694_mono__nat__linear__lb,axiom,
    ! [F4: nat > nat,M2: nat,K3: nat] :
      ( ! [M6: nat,N5: nat] :
          ( ( ord_less_nat @ M6 @ N5 )
         => ( ord_less_nat @ ( F4 @ M6 ) @ ( F4 @ N5 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F4 @ M2 ) @ K3 ) @ ( F4 @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_3695_Suc__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M2 ) @ ( times_times_nat @ ( suc @ K3 ) @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_3696_less__diff__conv,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ).

% less_diff_conv
thf(fact_3697_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_3698_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N7: nat] :
      ( ( ord_less_nat @ A2 @ A )
     => ( ( ord_less_nat @ B2 @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( times_times_nat @ A @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_3699_mlex__fst__decrI,axiom,
    ! [A2: nat,A7: nat,B2: nat,N7: nat,B6: nat] :
      ( ( ord_less_nat @ A2 @ A7 )
     => ( ( ord_less_nat @ B2 @ N7 )
       => ( ( ord_less_nat @ B6 @ N7 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A7 @ N7 ) @ B6 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_3700_mlex__snd__decrI,axiom,
    ! [A2: nat,A7: nat,B2: nat,B6: nat,N7: nat] :
      ( ( A2 = A7 )
     => ( ( ord_less_nat @ B2 @ B6 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A7 @ N7 ) @ B6 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_3701_mlex__leI,axiom,
    ! [A2: nat,A7: nat,B2: nat,B6: nat,N7: nat] :
      ( ( ord_less_eq_nat @ A2 @ A7 )
     => ( ( ord_less_eq_nat @ B2 @ B6 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A7 @ N7 ) @ B6 ) ) ) ) ).

% mlex_leI
thf(fact_3702_le__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3703_le__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3704_le__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3705_le__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3706_le__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3707_le__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3708_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3709_less__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3710_less__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3711_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3712_less__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3713_less__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3714_square__diff__one__factored,axiom,
    ! [X: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).

% square_diff_one_factored
thf(fact_3715_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_3716_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3717_less__diff__conv2,axiom,
    ! [K3: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K3 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ) ).

% less_diff_conv2
thf(fact_3718_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_3719_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3720_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3721_rev__nth,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
     => ( ( nth_c @ ( rev_c @ Xs ) @ N )
        = ( nth_c @ Xs @ ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3722_rev__nth,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
     => ( ( nth_d @ ( rev_d @ Xs ) @ N )
        = ( nth_d @ Xs @ ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3723_rev__nth,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3724_rev__nth,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rev_int @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3725_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_3726_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3727_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_3728_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3729_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3730_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_Code_integer,X: code_integer] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s3445333598471063425nteger @ L ) )
         => ( ( sorted710888440204495920nteger @ ord_le3102999989581377725nteger @ L )
           => ( ( ord_le3102999989581377725nteger @ ( nth_Code_integer @ L @ I ) @ X )
             => ( ( ord_le6747313008572928689nteger @ X @ ( nth_Code_integer @ L @ J ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I @ K )
                     => ( ( ord_less_nat @ K @ J )
                       => ( ( ord_le3102999989581377725nteger @ ( nth_Code_integer @ L @ K ) @ X )
                         => ~ ( ord_le6747313008572928689nteger @ X @ ( nth_Code_integer @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_3731_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,X: num] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_num @ L ) )
         => ( ( sorted_wrt_num @ ord_less_eq_num @ L )
           => ( ( ord_less_eq_num @ ( nth_num @ L @ I ) @ X )
             => ( ( ord_less_num @ X @ ( nth_num @ L @ J ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I @ K )
                     => ( ( ord_less_nat @ K @ J )
                       => ( ( ord_less_eq_num @ ( nth_num @ L @ K ) @ X )
                         => ~ ( ord_less_num @ X @ ( nth_num @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_3732_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ L ) )
         => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
           => ( ( ord_less_eq_nat @ ( nth_nat @ L @ I ) @ X )
             => ( ( ord_less_nat @ X @ ( nth_nat @ L @ J ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I @ K )
                     => ( ( ord_less_nat @ K @ J )
                       => ( ( ord_less_eq_nat @ ( nth_nat @ L @ K ) @ X )
                         => ~ ( ord_less_nat @ X @ ( nth_nat @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_3733_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ L ) )
         => ( ( sorted_wrt_int @ ord_less_eq_int @ L )
           => ( ( ord_less_eq_int @ ( nth_int @ L @ I ) @ X )
             => ( ( ord_less_int @ X @ ( nth_int @ L @ J ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I @ K )
                     => ( ( ord_less_nat @ K @ J )
                       => ( ( ord_less_eq_int @ ( nth_int @ L @ K ) @ X )
                         => ~ ( ord_less_int @ X @ ( nth_int @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_3734_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3735_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3736_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3737_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3738_crossproduct__noteq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) )
       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ D2 ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_3739_crossproduct__noteq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A2 @ D2 ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_3740_crossproduct__noteq,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_3741_crossproduct__noteq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_3742_nth__step__trancl,axiom,
    ! [Xs: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_s8736152011456118867at_nat @ Xs ) @ one_one_nat ) )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ ( suc @ N5 ) ) @ ( nth_se5679702624988424552at_nat @ Xs @ N5 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_s8736152011456118867at_nat @ Xs ) )
       => ( ( ord_less_nat @ M2 @ N )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ N ) @ ( nth_se5679702624988424552at_nat @ Xs @ M2 ) ) @ ( transi2703068831062848130at_nat @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_3743_nth__step__trancl,axiom,
    ! [Xs: list_c,R: set_Product_prod_c_c,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_c @ Xs ) @ one_one_nat ) )
         => ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ ( nth_c @ Xs @ ( suc @ N5 ) ) @ ( nth_c @ Xs @ N5 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_c @ Xs ) )
       => ( ( ord_less_nat @ M2 @ N )
         => ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ ( nth_c @ Xs @ N ) @ ( nth_c @ Xs @ M2 ) ) @ ( transitive_trancl_c @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_3744_nth__step__trancl,axiom,
    ! [Xs: list_d,R: set_Product_prod_d_d,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_d @ Xs ) @ one_one_nat ) )
         => ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ ( nth_d @ Xs @ ( suc @ N5 ) ) @ ( nth_d @ Xs @ N5 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_d @ Xs ) )
       => ( ( ord_less_nat @ M2 @ N )
         => ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ ( nth_d @ Xs @ N ) @ ( nth_d @ Xs @ M2 ) ) @ ( transitive_trancl_d @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_3745_nth__step__trancl,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ ( suc @ N5 ) ) @ ( nth_int @ Xs @ N5 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ M2 @ N )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ ( nth_int @ Xs @ M2 ) ) @ ( transi6261509568448316235cl_int @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_3746_nth__step__trancl,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( suc @ N5 ) ) @ ( nth_nat @ Xs @ N5 ) ) @ R ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ M2 @ N )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ ( nth_nat @ Xs @ M2 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% nth_step_trancl
thf(fact_3747_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_3748_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_3749_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_3750_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_3751_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_3752_add_Oright__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% add.right_neutral
thf(fact_3753_add_Oright__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_3754_add_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.right_neutral
thf(fact_3755_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_3756_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_3757_double__zero__sym,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_3758_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_3759_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_3760_add__cancel__left__left,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_3761_add__cancel__left__left,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
thf(fact_3762_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_3763_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_3764_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_3765_add__cancel__left__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_3766_add__cancel__left__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
thf(fact_3767_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_3768_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_3769_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_3770_add__cancel__right__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_3771_add__cancel__right__left,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2
        = ( plus_p7104986032573967614at_nat @ B2 @ A2 ) )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
thf(fact_3772_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ B2 @ A2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_3773_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_3774_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ B2 @ A2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_3775_add__cancel__right__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_3776_add__cancel__right__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2
        = ( plus_p7104986032573967614at_nat @ A2 @ B2 ) )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
thf(fact_3777_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ A2 @ B2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_3778_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_3779_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ A2 @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_3780_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_3781_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_3782_add__0,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% add_0
thf(fact_3783_add__0,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_3784_add__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add_0
thf(fact_3785_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_3786_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_3787_mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ C )
        = ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_3788_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_3789_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ C )
        = ( times_times_nat @ B2 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_3790_mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ( times_times_int @ A2 @ C )
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_3791_mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A2 )
        = ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_3792_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_3793_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ( times_times_nat @ C @ A2 )
        = ( times_times_nat @ C @ B2 ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_3794_mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_3795_mult__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
      = ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_3796_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_3797_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_3798_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_3799_mult__zero__right,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_3800_mult__zero__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_3801_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_3802_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_3803_mult__zero__left,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_3804_mult__zero__left,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_3805_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_3806_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_3807_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3808_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ A2 )
      = zero_z1048942125864253310at_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3809_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3810_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3811_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3812_diff__zero,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% diff_zero
thf(fact_3813_diff__zero,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% diff_zero
thf(fact_3814_diff__zero,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_zero
thf(fact_3815_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_3816_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_3817_zero__diff,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = zero_z1048942125864253310at_nat ) ).

% zero_diff
thf(fact_3818_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3819_diff__0__right,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% diff_0_right
thf(fact_3820_diff__0__right,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_0_right
thf(fact_3821_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_3822_diff__self,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_3823_diff__self,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3824_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_3825_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_3826_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_3827_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_3828_trancl__single,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( transi2703068831062848130at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ bot_bo228742789529271731at_nat ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ bot_bo228742789529271731at_nat ) ) ).

% trancl_single
thf(fact_3829_trancl__single,axiom,
    ! [A2: nat,B2: nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ bot_bo2099793752762293965at_nat ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ bot_bo2099793752762293965at_nat ) ) ).

% trancl_single
thf(fact_3830_add__le__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_3831_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_3832_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_3833_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_3834_add__le__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_3835_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_3836_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_3837_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_3838_le__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_3839_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_3840_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_3841_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_3842_le__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_3843_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_3844_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_3845_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_3846_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3847_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3848_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3849_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3850_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3851_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3852_diff__ge__0__iff__ge,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_3853_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_3854_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_3855_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3856_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3857_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3858_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3859_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3860_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3861_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3862_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3863_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3864_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_3865_less__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_3866_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_3867_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_3868_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_3869_less__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_3870_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_3871_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_3872_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_3873_add__less__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_3874_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_3875_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_3876_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_3877_add__less__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_3878_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_3879_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_3880_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_3881_diff__gt__0__iff__gt,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_3882_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_3883_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3884_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3885_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3886_mult__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_3887_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_3888_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_3889_mult__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A2 )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_3890_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_3891_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_3892_mult__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_3893_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_3894_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_3895_mult__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ C )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_3896_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_3897_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ( times_times_int @ A2 @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_3898_diff__add__zero,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) )
      = zero_z1048942125864253310at_nat ) ).

% diff_add_zero
thf(fact_3899_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3900_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_3901_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3902_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3903_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_3904_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_3905_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_3906_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_3907_nat__mult__less__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K3 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_3908_mult__less__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K3 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% mult_less_cancel2
thf(fact_3909_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_3910_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_3911_slice__complete,axiom,
    ! [Xs: list_c] :
      ( ( slice_c @ zero_zero_nat @ ( size_size_list_c @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3912_slice__complete,axiom,
    ! [Xs: list_d] :
      ( ( slice_d @ zero_zero_nat @ ( size_size_list_d @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3913_slice__complete,axiom,
    ! [Xs: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3914_slice__complete,axiom,
    ! [Xs: list_int] :
      ( ( slice_int @ zero_zero_nat @ ( size_size_list_int @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3915_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_3916_nat__mult__le__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_3917_mult__le__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% mult_le_cancel2
thf(fact_3918_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_3919_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% assn_aci(8)
thf(fact_3920_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% assn_aci(7)
thf(fact_3921_assn__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y5: assn] : ( sup_sup_assn @ Y5 @ X2 ) ) ) ).

% assn_aci(5)
thf(fact_3922_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(32)
thf(fact_3923_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(15)
thf(fact_3924_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_3925_zero__reorient,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = X )
      = ( X = zero_z1048942125864253310at_nat ) ) ).

% zero_reorient
thf(fact_3926_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_3927_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_3928_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_3929_trancl__mono__mp,axiom,
    ! [U2: set_Pr4329608150637261639at_nat,V: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ U2 @ V )
     => ( ( member8757157785044589968at_nat @ X @ ( transi2703068831062848130at_nat @ U2 ) )
       => ( member8757157785044589968at_nat @ X @ ( transi2703068831062848130at_nat @ V ) ) ) ) ).

% trancl_mono_mp
thf(fact_3930_trancl__mono__mp,axiom,
    ! [U2: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ U2 @ V )
     => ( ( member8440522571783428010at_nat @ X @ ( transi6264000038957366511cl_nat @ U2 ) )
       => ( member8440522571783428010at_nat @ X @ ( transi6264000038957366511cl_nat @ V ) ) ) ) ).

% trancl_mono_mp
thf(fact_3931_trancl__sub,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ R @ ( transi6264000038957366511cl_nat @ R ) ) ).

% trancl_sub
thf(fact_3932_trancl__sub__insert__trancl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( transi2703068831062848130at_nat @ R ) @ ( transi2703068831062848130at_nat @ ( insert9069300056098147895at_nat @ X @ R ) ) ) ).

% trancl_sub_insert_trancl
thf(fact_3933_trancl__sub__insert__trancl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( transi6264000038957366511cl_nat @ R ) @ ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ X @ R ) ) ) ).

% trancl_sub_insert_trancl
thf(fact_3934_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_3935_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_3936_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_3937_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_3938_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_3939_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_3940_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_numeral_extra(3)
thf(fact_3941_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_3942_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_3943_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3944_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3945_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3946_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3947_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3948_add_Ocomm__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% add.comm_neutral
thf(fact_3949_add_Ocomm__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_3950_add_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.comm_neutral
thf(fact_3951_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_3952_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_3953_add_Ogroup__left__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3954_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3955_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3956_mult__right__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A2 @ C )
          = ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_3957_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C )
          = ( times_times_rat @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_3958_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C )
          = ( times_times_nat @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_3959_mult__right__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C )
          = ( times_times_int @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_3960_mult__left__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C @ A2 )
          = ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_3961_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A2 )
          = ( times_times_rat @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_3962_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A2 )
          = ( times_times_nat @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_3963_mult__left__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A2 )
          = ( times_times_int @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_3964_no__zero__divisors,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B2 != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A2 @ B2 )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_3965_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( times_times_rat @ A2 @ B2 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_3966_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A2 @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_3967_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A2 @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_3968_divisors__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_3969_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_3970_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_3971_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_3972_mult__not__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
       != zero_z3403309356797280102nteger )
     => ( ( A2 != zero_z3403309356797280102nteger )
        & ( B2 != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_3973_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
       != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_3974_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
       != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_3975_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
       != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_3976_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_3977_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_3978_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_3979_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_3980_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: code_integer,Z2: code_integer] : Y4 = Z2 )
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( minus_8373710615458151222nteger @ A3 @ B3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3981_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
    = ( ^ [A3: rat,B3: rat] :
          ( ( minus_minus_rat @ A3 @ B3 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3982_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3983_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ~ ( P @ N5 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N5 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_3984_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_3985_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_3986_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_3987_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_3988_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_3989_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_3990_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ R2 @ C ) )
         != ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3991_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B2 @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3992_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3993_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B2: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3994_nat__mult__eq__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( ( times_times_nat @ K3 @ M2 )
          = ( times_times_nat @ K3 @ N ) )
        = ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_3995_nat__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_3996_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3997_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3998_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3999_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_4000_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_4001_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_4002_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_4003_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_4004_add__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_4005_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_4006_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_4007_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_4008_add__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_4009_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_4010_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_4011_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_4012_add__increasing2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
       => ( ord_le3102999989581377725nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_4013_add__increasing2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_4014_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_4015_add__increasing2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_4016_add__decreasing2,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_4017_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_4018_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_4019_add__decreasing2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_4020_add__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C )
       => ( ord_le3102999989581377725nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_4021_add__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_4022_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_4023_add__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_4024_add__decreasing,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C @ B2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_4025_add__decreasing,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_4026_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_4027_add__decreasing,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_4028_mult__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_4029_mult__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_4030_mult__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_4031_mult__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_4032_mult__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4033_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4034_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4035_mult__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4036_zero__le__square,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4037_zero__le__square,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4038_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4039_split__mult__pos__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_4040_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_4041_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_4042_mult__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4043_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4044_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4045_mult__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4046_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4047_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4048_mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_4049_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_4050_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_4051_mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_4052_mult__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4053_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4054_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4055_mult__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4056_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4057_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4058_mult__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4059_mult__le__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4060_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4061_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4062_split__mult__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_4063_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_4064_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_4065_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_4066_mult__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4067_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4068_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4069_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4070_mult__nonneg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4071_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4072_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4073_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4074_mult__nonpos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4075_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4076_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4077_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4078_mult__nonneg__nonpos2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4079_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4080_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4081_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4082_zero__le__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4083_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4084_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4085_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4086_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4087_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4088_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4089_zero__less__one__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one_class.zero_le_one
thf(fact_4090_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_4091_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_4092_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_4093_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_4094_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_4095_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_4096_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_4097_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_4098_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_4099_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_4100_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_4101_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_4102_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4103_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_4104_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_4105_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ X @ zero_z3403309356797280102nteger )
        | ( ord_le6747313008572928689nteger @ Y @ zero_z3403309356797280102nteger ) ) ) ).

% add_less_zeroD
thf(fact_4106_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_4107_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_4108_add__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_4109_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_4110_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_4111_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_4112_add__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_4113_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_4114_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_4115_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ! [C4: nat] :
            ( ( B2
              = ( plus_plus_nat @ A2 @ C4 ) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_4116_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_4117_pos__add__strict,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_4118_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_4119_pos__add__strict,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_4120_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_4121_mult__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_4122_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_4123_not__square__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_4124_not__square__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_4125_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_4126_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4127_mult__less__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4128_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4129_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_4130_mult__neg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_4131_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_4132_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_4133_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_4134_mult__pos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_4135_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_4136_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_4137_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_4138_mult__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_4139_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_4140_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_4141_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_4142_mult__pos__neg2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_4143_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_4144_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_4145_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4146_zero__less__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4147_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4148_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_4149_zero__less__mult__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_4150_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_4151_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_4152_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4153_zero__less__mult__pos2,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4154_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4155_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4156_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4157_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4158_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4159_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4160_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4161_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4162_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4163_mult__strict__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4164_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4165_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4166_mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4167_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4168_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4169_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4170_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4171_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4172_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4173_mult__strict__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4174_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4175_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4176_mult__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4177_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4178_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4179_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4180_mult__less__cancel__right__disj,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4181_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4182_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4183_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4184_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4185_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4186_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_4187_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_4188_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_4189_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_4190_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_4191_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_4192_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_4193_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_4194_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_4195_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_4196_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_4197_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_4198_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4199_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_4200_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_4201_nat__mult__le__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_4202_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_4203_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M: nat] :
            ( N
            = ( suc @ M ) ) ) ) ).

% gr0_conv_Suc
thf(fact_4204_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_4205_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% gr0_implies_Suc
thf(fact_4206_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_4207_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N5 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_4208_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N5 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_4209_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I6: nat] :
                ( ( ord_less_nat @ I6 @ K )
               => ~ ( P @ I6 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_4210_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I @ K )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_4211_Suc__to__right,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( suc @ N )
        = M2 )
     => ( N
        = ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_4212_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_4213_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).

% mult_less_mono1
thf(fact_4214_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_4215_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_4216_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_4217_add__neg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_4218_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_4219_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_4220_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_4221_add__nonneg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_4222_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_4223_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_4224_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_4225_add__nonpos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_4226_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_4227_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_4228_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_4229_add__pos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_4230_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_4231_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_4232_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_4233_add__strict__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_4234_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_4235_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_4236_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_4237_add__strict__increasing2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_4238_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_4239_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_4240_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4241_mult__le__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4242_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4243_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4244_mult__le__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4245_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4246_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4247_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4248_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4249_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4250_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4251_mult__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4252_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4253_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4254_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4255_mult__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4256_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4257_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4258_mult__right__less__imp__less,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4259_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4260_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4261_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4262_mult__strict__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4263_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4264_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4265_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4266_mult__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4267_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4268_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4269_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4270_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4271_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4272_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4273_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4274_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4275_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4276_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4277_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4278_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4279_mult__right__le__imp__le,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4280_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4281_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4282_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4283_mult__le__less__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4284_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4285_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4286_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4287_mult__less__le__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4288_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4289_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4290_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_4291_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_4292_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_4293_mult__left__le,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4294_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4295_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4296_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4297_mult__le__one,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_4298_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_4299_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_4300_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_4301_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4302_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4303_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4304_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4305_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4306_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4307_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_4308_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_squares_lt_zero
thf(fact_4309_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_4310_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_4311_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_4312_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_4313_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_4314_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I6: nat] :
                ( ( ord_less_eq_nat @ I6 @ K )
               => ~ ( P @ I6 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_4315_length__pos__if__in__set,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1588839187904960729at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4316_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4317_length__pos__if__in__set,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8908918004648414271_nat_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4318_length__pos__if__in__set,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4319_length__pos__if__in__set,axiom,
    ! [X: c,Xs: list_c] :
      ( ( member_c @ X @ ( set_c2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_c @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4320_length__pos__if__in__set,axiom,
    ! [X: d,Xs: list_d] :
      ( ( member_d @ X @ ( set_d2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_d @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4321_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4322_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4323_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_4324_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_4325_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_4326_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% one_less_mult
thf(fact_4327_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( P @ N5 )
               => ( P @ ( suc @ N5 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_4328_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ( ( ord_less_nat @ A2 @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A2
              = ( plus_plus_nat @ B2 @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_4329_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B2 @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4330_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4331_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4332_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4333_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4334_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4335_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4336_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4337_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4338_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4339_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4340_mult__le__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4341_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4342_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4343_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4344_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4345_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4346_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4347_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4348_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4349_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4350_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4351_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4352_mult__less__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4353_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4354_convex__bound__le,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A2 )
     => ( ( ord_le3102999989581377725nteger @ Y @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4355_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_eq_rat @ X @ A2 )
     => ( ( ord_less_eq_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4356_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4357_nz__le__conv__less,axiom,
    ! [K3: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( ord_less_eq_nat @ K3 @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ K3 @ ( suc @ zero_zero_nat ) ) @ M2 ) ) ) ).

% nz_le_conv_less
thf(fact_4358_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_4359_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_4360_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_rat @ X @ A2 )
     => ( ( ord_less_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4361_convex__bound__lt,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A2 )
     => ( ( ord_le6747313008572928689nteger @ Y @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4362_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V2: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ( ord_less_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4363_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_4364_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4365_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4366_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4367_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4368_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4369_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4370_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4371_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4372_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4373_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4374_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4375_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4376_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_4377_divides__aux__eq,axiom,
    ! [Q6: code_integer,R2: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q6 @ R2 ) )
      = ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_4378_divides__aux__eq,axiom,
    ! [Q6: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q6 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_4379_divides__aux__eq,axiom,
    ! [Q6: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_4380_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4381_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4382_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4383_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4384_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4385_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4386_converse__trancl__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ B2 ) @ R2 )
           => ( P @ Y2 ) )
       => ( ! [Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ Z4 ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z4 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
               => ( ( P @ Z4 )
                 => ( P @ Y2 ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% converse_trancl_induct
thf(fact_4387_converse__trancl__induct,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B2 ) @ R2 )
           => ( P @ Y2 ) )
       => ( ! [Y2: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z4 ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z4 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
               => ( ( P @ Z4 )
                 => ( P @ Y2 ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% converse_trancl_induct
thf(fact_4388_trancl__trans__induct,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ R2 )
           => ( P @ X3 @ Y2 ) )
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
             => ( ( P @ X3 @ Y2 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ Z4 ) @ ( transi2703068831062848130at_nat @ R2 ) )
                 => ( ( P @ Y2 @ Z4 )
                   => ( P @ X3 @ Z4 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_4389_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [X3: nat,Y2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
           => ( P @ X3 @ Y2 ) )
       => ( ! [X3: nat,Y2: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( P @ X3 @ Y2 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                 => ( ( P @ Y2 @ Z4 )
                   => ( P @ X3 @ Z4 ) ) ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_4390_trancl__into__trancl2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ ( transi2703068831062848130at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ C ) @ ( transi2703068831062848130at_nat @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_4391_trancl__into__trancl2,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_4392_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ R2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ C ) @ ( transi2703068831062848130at_nat @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_4393_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_4394_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] :
          ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_4395_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_4396_converse__tranclE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ R2 )
       => ~ ! [Y2: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y2 ) @ R2 )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ Z ) @ ( transi2703068831062848130at_nat @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_4397_converse__tranclE,axiom,
    ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
       => ~ ! [Y2: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_4398_r__r__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ R )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ C ) @ ( transi2703068831062848130at_nat @ R ) ) ) ) ).

% r_r_into_trancl
thf(fact_4399_r__r__into__trancl,axiom,
    ! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% r_r_into_trancl
thf(fact_4400_trancl__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ Y2 ) @ R2 )
           => ( P @ Y2 ) )
       => ( ! [Y2: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ Y2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ Z4 ) @ R2 )
               => ( ( P @ Y2 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% trancl_induct
thf(fact_4401_trancl__induct,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ R2 )
           => ( P @ Y2 ) )
       => ( ! [Y2: nat,Z4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z4 ) @ R2 )
               => ( ( P @ Y2 )
                 => ( P @ Z4 ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% trancl_induct
thf(fact_4402_trancl__induct2,axiom,
    ! [Ax: produc6241069584506657477e_term > option6357759511663192854e_term,Ay: produc8923325533196201883nteger,Bx: produc6241069584506657477e_term > option6357759511663192854e_term,By: produc8923325533196201883nteger,R2: set_Pr1722956316086891245nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ Bx @ By ) ) @ ( transi6228740973808232501nteger @ R2 ) )
     => ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
            ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A4 @ B4 ) ) @ R2 )
           => ( P @ A4 @ B4 ) )
       => ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger,Aa: produc6241069584506657477e_term > option6357759511663192854e_term,Ba: produc8923325533196201883nteger] :
              ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ Ax @ Ay ) @ ( produc8603105652947943368nteger @ A4 @ B4 ) ) @ ( transi6228740973808232501nteger @ R2 ) )
             => ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A4 @ B4 ) @ ( produc8603105652947943368nteger @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_4403_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
     => ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
            ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B4 ) ) @ R2 )
           => ( P @ A4 @ B4 ) )
       => ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
              ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B4 ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
             => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B4 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_4404_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
     => ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
            ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B4 ) ) @ R2 )
           => ( P @ A4 @ B4 ) )
       => ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
              ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B4 ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
             => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B4 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_4405_trancl__induct2,axiom,
    ! [Ax: produc8551481072490612790e_term > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: produc8551481072490612790e_term > option6357759511663192854e_term,By: product_prod_int_int,R2: set_Pr4708930517165415495nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ Bx @ By ) ) @ ( transi7434818091516421410nt_int @ R2 ) )
     => ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A4 @ B4 ) ) @ R2 )
           => ( P @ A4 @ B4 ) )
       => ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int,Aa: produc8551481072490612790e_term > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ Ax @ Ay ) @ ( produc5700946648718959541nt_int @ A4 @ B4 ) ) @ ( transi7434818091516421410nt_int @ R2 ) )
             => ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A4 @ B4 ) @ ( produc5700946648718959541nt_int @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_4406_trancl__induct2,axiom,
    ! [Ax: int > option6357759511663192854e_term,Ay: product_prod_int_int,Bx: int > option6357759511663192854e_term,By: product_prod_int_int,R2: set_Pr8634505666381077339nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ Bx @ By ) ) @ ( transi4533545184864474924nt_int @ R2 ) )
     => ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A4 @ B4 ) ) @ R2 )
           => ( P @ A4 @ B4 ) )
       => ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int,Aa: int > option6357759511663192854e_term,Ba: product_prod_int_int] :
              ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ Ax @ Ay ) @ ( produc4305682042979456191nt_int @ A4 @ B4 ) ) @ ( transi4533545184864474924nt_int @ R2 ) )
             => ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A4 @ B4 ) @ ( produc4305682042979456191nt_int @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A4 @ B4 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_4407_trancl_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ R2 )
       => ~ ! [B4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ B4 ) @ ( transi2703068831062848130at_nat @ R2 ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_4408_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
       => ~ ! [B4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A22 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_4409_trancl_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( transi2703068831062848130at_nat @ R2 ) )
      = ( ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = B3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ R2 ) )
        | ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C5: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A3 )
            & ( A22 = C5 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ ( transi2703068831062848130at_nat @ R2 ) )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ C5 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_4410_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
      = ( ? [A3: nat,B3: nat] :
            ( ( A1 = A3 )
            & ( A22 = B3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R2 ) )
        | ? [A3: nat,B3: nat,C5: nat] :
            ( ( A1 = A3 )
            & ( A22 = C5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C5 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_4411_trancl_Or__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_4412_trancl_Or__into__trancl,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_4413_tranclE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 )
       => ~ ! [C4: set_Pr1261947904930325089at_nat] :
              ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ C4 ) @ ( transi2703068831062848130at_nat @ R2 ) )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C4 @ B2 ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_4414_tranclE,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
       => ~ ! [C4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C4 @ B2 ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_4415_trancl__trans,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transi2703068831062848130at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ ( transi2703068831062848130at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ ( transi2703068831062848130at_nat @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_4416_trancl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_4417_nonempty__has__size,axiom,
    ! [S: multis2468970476368604999at_nat] :
      ( ( S != zero_z1048942125864253310at_nat )
      = ( ord_less_nat @ zero_zero_nat @ ( size_s8510653225128441779at_nat @ S ) ) ) ).

% nonempty_has_size
thf(fact_4418_bijective__Empty,axiom,
    biject3388518990857851981at_nat @ bot_bo228742789529271731at_nat ).

% bijective_Empty
thf(fact_4419_bijective__Empty,axiom,
    bijective_nat_nat @ bot_bo2099793752762293965at_nat ).

% bijective_Empty
thf(fact_4420_total__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( total_7212417256332265445at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% total_on_singleton
thf(fact_4421_total__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( total_1513685709984341995at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% total_on_singleton
thf(fact_4422_total__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( total_3592101749530773125at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% total_on_singleton
thf(fact_4423_total__on__singleton,axiom,
    ! [X: $o] : ( total_on_o @ ( insert_o2 @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).

% total_on_singleton
thf(fact_4424_total__on__singleton,axiom,
    ! [X: nat] : ( total_on_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% total_on_singleton
thf(fact_4425_total__on__singleton,axiom,
    ! [X: int] : ( total_on_int @ ( insert_int2 @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).

% total_on_singleton
thf(fact_4426_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4427_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4428_nth__Cons__pos,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
        = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4429_Field__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A2 @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_4430_Field__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A2 @ B2 ) @ R2 ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_4431_Field__insert,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_4432_Field__insert,axiom,
    ! [A2: $o,B2: $o,R2: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R2 ) )
      = ( sup_sup_set_o @ ( insert_o2 @ A2 @ ( insert_o2 @ B2 @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).

% Field_insert
thf(fact_4433_Field__insert,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) )
      = ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_4434_Field__insert,axiom,
    ! [A2: int,B2: int,R2: set_Pr958786334691620121nt_int] :
      ( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) )
      = ( sup_sup_set_int @ ( insert_int2 @ A2 @ ( insert_int2 @ B2 @ bot_bot_set_int ) ) @ ( field_int @ R2 ) ) ) ).

% Field_insert
thf(fact_4435_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_c,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_c @ Xs ) )
     => ( ( nth_Pr2476257089995772766_nat_c @ ( enumerate_c @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_c @ ( plus_plus_nat @ N @ M2 ) @ ( nth_c @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_4436_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_d,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_d @ Xs ) )
     => ( ( nth_Pr2476257094299001567_nat_d @ ( enumerate_d @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_d @ ( plus_plus_nat @ N @ M2 ) @ ( nth_d @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_4437_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_4438_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_int,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M2 ) @ ( nth_int @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_4439_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ? [As13: set_nat,As24: set_nat] :
                    ( ( As
                      = ( sup_sup_set_nat @ As13 @ As24 ) )
                    & ( ( inf_inf_set_nat @ As13 @ As24 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_4440_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ~ ? [As12: set_nat,As23: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As12 @ As23 ) )
                      & ( ( inf_inf_set_nat @ As12 @ As23 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_4441_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
      ( ( ( cons_int @ X21 @ X222 )
        = ( cons_int @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_4442_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_4443_list_Oinject,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn,Y21: produc6575502325842934193n_assn,Y222: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X21 @ X222 )
        = ( cons_P2971678138204555879n_assn @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_4444_Field__Un,axiom,
    ! [R2: set_Product_prod_o_o,S3: set_Product_prod_o_o] :
      ( ( field_o @ ( sup_su5769328420594410459od_o_o @ R2 @ S3 ) )
      = ( sup_sup_set_o @ ( field_o @ R2 ) @ ( field_o @ S3 ) ) ) ).

% Field_Un
thf(fact_4445_Field__Un,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) )
      = ( sup_sup_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S3 ) ) ) ).

% Field_Un
thf(fact_4446_Field__Un,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,S3: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( sup_su6664651903789602459at_nat @ R2 @ S3 ) )
      = ( sup_su5525570899277871387at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ ( field_2862844103579723188at_nat @ S3 ) ) ) ).

% Field_Un
thf(fact_4447_Field__Un,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( sup_su5525570899277871387at_nat @ R2 @ S3 ) )
      = ( sup_su3642409539654194069at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ ( field_3145993188331922222at_nat @ S3 ) ) ) ).

% Field_Un
thf(fact_4448_list__ex__simps_I1_J,axiom,
    ! [P: int > $o,X: int,Xs: list_int] :
      ( ( list_ex_int @ P @ ( cons_int @ X @ Xs ) )
      = ( ( P @ X )
        | ( list_ex_int @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_4449_list__ex__simps_I1_J,axiom,
    ! [P: nat > $o,X: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P @ ( cons_nat @ X @ Xs ) )
      = ( ( P @ X )
        | ( list_ex_nat @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_4450_list__ex__simps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( list_e1087220904399170805n_assn @ P @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( ( P @ X )
        | ( list_e1087220904399170805n_assn @ P @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_4451_length__enumerate,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( size_s385972143775086793_nat_c @ ( enumerate_c @ N @ Xs ) )
      = ( size_size_list_c @ Xs ) ) ).

% length_enumerate
thf(fact_4452_length__enumerate,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( size_s457006183821432778_nat_d @ ( enumerate_d @ N @ Xs ) )
      = ( size_size_list_d @ Xs ) ) ).

% length_enumerate
thf(fact_4453_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_4454_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_enumerate
thf(fact_4455_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X222 ) )
      = ( insert8211810215607154385at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4456_list_Osimps_I15_J,axiom,
    ! [X21: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( cons_P3940603068885512221at_nat @ X21 @ X222 ) )
      = ( insert9069300056098147895at_nat @ X21 @ ( set_Pr3765526544606949372at_nat @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4457_list_Osimps_I15_J,axiom,
    ! [X21: $o,X222: list_o] :
      ( ( set_o2 @ ( cons_o @ X21 @ X222 ) )
      = ( insert_o2 @ X21 @ ( set_o2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4458_list_Osimps_I15_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( set_int2 @ ( cons_int @ X21 @ X222 ) )
      = ( insert_int2 @ X21 @ ( set_int2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4459_list_Osimps_I15_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
      = ( insert_nat2 @ X21 @ ( set_nat2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4460_list_Osimps_I15_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn @ ( cons_P2971678138204555879n_assn @ X21 @ X222 ) )
      = ( insert5290817439147925377n_assn @ X21 @ ( set_Pr1139785259514867910n_assn @ X222 ) ) ) ).

% list.simps(15)
thf(fact_4461_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_4462_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_4463_nth__Cons__0,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_4464_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_4465_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_4466_nth__Cons__Suc,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr1769885009046257848n_assn @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_4467_zip__Cons__Cons,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] :
      ( ( zip_assn_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) )
      = ( cons_P2971678138204555879n_assn @ ( produc118845697133431529n_assn @ X @ Y ) @ ( zip_assn_assn @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4468_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4469_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( zip_int_nat @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( zip_int_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4470_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( zip_nat_int @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( zip_nat_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4471_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4472_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_in7098856637274830732n_assn @ ( cons_int @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( cons_P8828502514829229636n_assn @ ( produc3313821056773121708n_assn @ X @ Y ) @ ( zip_in7098856637274830732n_assn @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4473_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_na5338819973521719528n_assn @ ( cons_nat @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( cons_P7068465851076118432n_assn @ ( produc1553784393020010504n_assn @ X @ Y ) @ ( zip_na5338819973521719528n_assn @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4474_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] :
      ( ( zip_Pr1015554993053626726sn_int @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P5155970366129889750sn_int @ ( produc6453891449406693510sn_int @ X @ Y ) @ ( zip_Pr1015554993053626726sn_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4475_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] :
      ( ( zip_Pr1018045463562677002sn_nat @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P110449348784310650sn_nat @ ( produc6456381919915743786sn_nat @ X @ Y ) @ ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4476_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_Pr8806919808394484215n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( cons_P9150528168557782045n_assn @ ( produc6128706242671166167n_assn @ X @ Y ) @ ( zip_Pr8806919808394484215n_assn @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_4477_Field__empty,axiom,
    ( ( field_2862844103579723188at_nat @ bot_bo7193250159860175667at_nat )
    = bot_bo228742789529271731at_nat ) ).

% Field_empty
thf(fact_4478_Field__empty,axiom,
    ( ( field_4706345496283612622at_nat @ bot_bo5327735625951526323at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Field_empty
thf(fact_4479_Field__empty,axiom,
    ( ( field_o @ bot_bo7073875226086086771od_o_o )
    = bot_bot_set_o ) ).

% Field_empty
thf(fact_4480_Field__empty,axiom,
    ( ( field_int @ bot_bo1796632182523588997nt_int )
    = bot_bot_set_int ) ).

% Field_empty
thf(fact_4481_Field__empty,axiom,
    ( ( field_3145993188331922222at_nat @ bot_bo228742789529271731at_nat )
    = bot_bo3083307316010499117at_nat ) ).

% Field_empty
thf(fact_4482_Field__empty,axiom,
    ( ( field_nat @ bot_bo2099793752762293965at_nat )
    = bot_bot_set_nat ) ).

% Field_empty
thf(fact_4483_not__in__set__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( insert4149901806602606979at_nat @ X @ Xs )
        = ( cons_P3940603068885512221at_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4484_not__in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X @ Xs )
        = ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4485_not__in__set__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( insert8021100735160035049_nat_o @ X @ Xs )
        = ( cons_P6219271836124797827_nat_o @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4486_not__in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( insert_int @ X @ Xs )
        = ( cons_int @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4487_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = ( cons_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4488_not__in__set__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( insert3246601298802261197n_assn @ X @ Xs )
        = ( cons_P2971678138204555879n_assn @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_4489_Cons__listrel1__Cons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre8180072825634266816at_nat @ R2 ) )
      = ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( listre8180072825634266816at_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_4490_Cons__listrel1__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R2 ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_4491_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_4492_Cons__listrel1__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre4270255209692100112n_assn @ R2 ) )
      = ( ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X @ Y ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ Ys ) @ ( listre4270255209692100112n_assn @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_4493_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( enumerate_int @ N @ ( cons_int @ X @ Xs ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_4494_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_4495_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( enumer9200277086812776726n_assn @ N @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( cons_P7068465851076118432n_assn @ ( produc1553784393020010504n_assn @ N @ X ) @ ( enumer9200277086812776726n_assn @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_4496_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] :
      ( ( cons_int @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_4497_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_4498_not__Cons__self2,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( cons_P2971678138204555879n_assn @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_4499_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N22: int,R22: list_int] :
      ( ( ( cons_int @ N1 @ R1 )
        = ( cons_int @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_4500_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N22: nat,R22: list_nat] :
      ( ( ( cons_nat @ N1 @ R1 )
        = ( cons_nat @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_4501_list__tail__coinc,axiom,
    ! [N1: produc6575502325842934193n_assn,R1: list_P8527749157015355191n_assn,N22: produc6575502325842934193n_assn,R22: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ N1 @ R1 )
        = ( cons_P2971678138204555879n_assn @ N22 @ R22 ) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
thf(fact_4502_union__diff__assoc,axiom,
    ! [C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( ( minus_4286766774447292334at_nat @ C2 @ B )
        = zero_z1048942125864253310at_nat )
     => ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ C2 )
        = ( plus_p7104986032573967614at_nat @ A @ ( minus_4286766774447292334at_nat @ B @ C2 ) ) ) ) ).

% union_diff_assoc
thf(fact_4503_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y2: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4504_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys )
        = ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y2: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_nat @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_int_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4505_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( ( zip_nat_int @ Xs @ Ys )
        = ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y2: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_int @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_nat_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4506_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y2: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4507_zip__eq__ConsE,axiom,
    ! [Xs: list_assn,Ys: list_assn,Xy: produc6575502325842934193n_assn,Xys: list_P8527749157015355191n_assn] :
      ( ( ( zip_assn_assn @ Xs @ Ys )
        = ( cons_P2971678138204555879n_assn @ Xy @ Xys ) )
     => ~ ! [X3: assn,Xs5: list_assn] :
            ( ( Xs
              = ( cons_assn @ X3 @ Xs5 ) )
           => ! [Y2: assn,Ys4: list_assn] :
                ( ( Ys
                  = ( cons_assn @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc118845697133431529n_assn @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_assn_assn @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4508_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn,Xy: produc3167338625816234618n_assn,Xys: list_P1495281457390866698n_assn] :
      ( ( ( zip_in7098856637274830732n_assn @ Xs @ Ys )
        = ( cons_P8828502514829229636n_assn @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs5: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs5 ) )
           => ! [Y2: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
                ( ( Ys
                  = ( cons_P2971678138204555879n_assn @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc3313821056773121708n_assn @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_in7098856637274830732n_assn @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4509_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn,Xy: produc1407301962063123414n_assn,Xys: list_P7711101519850957926n_assn] :
      ( ( ( zip_na5338819973521719528n_assn @ Xs @ Ys )
        = ( cons_P7068465851076118432n_assn @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs5 ) )
           => ! [Y2: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
                ( ( Ys
                  = ( cons_P2971678138204555879n_assn @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc1553784393020010504n_assn @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_na5338819973521719528n_assn @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4510_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int,Xy: produc8718178513971670540sn_int,Xys: list_P421884338359569308sn_int] :
      ( ( ( zip_Pr1015554993053626726sn_int @ Xs @ Ys )
        = ( cons_P5155970366129889750sn_int @ Xy @ Xys ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs5 ) )
           => ! [Y2: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6453891449406693510sn_int @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_Pr1015554993053626726sn_int @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4511_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat,Xy: produc3672657496626091440sn_nat,Xys: list_P2911967483291296064sn_nat] :
      ( ( ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys )
        = ( cons_P110449348784310650sn_nat @ Xy @ Xys ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs5 ) )
           => ! [Y2: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6456381919915743786sn_nat @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_Pr1018045463562677002sn_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4512_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Xy: produc9053633027152450663n_assn,Xys: list_P28140577958321773n_assn] :
      ( ( ( zip_Pr8806919808394484215n_assn @ Xs @ Ys )
        = ( cons_P9150528168557782045n_assn @ Xy @ Xys ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs5 ) )
           => ! [Y2: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
                ( ( Ys
                  = ( cons_P2971678138204555879n_assn @ Y2 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6128706242671166167n_assn @ X3 @ Y2 ) )
                 => ( Xys
                   != ( zip_Pr8806919808394484215n_assn @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_4513_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R ) ) ) ).

% FieldI1
thf(fact_4514_FieldI1,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R )
     => ( member8440522571783428010at_nat @ I @ ( field_4706345496283612622at_nat @ R ) ) ) ).

% FieldI1
thf(fact_4515_FieldI1,axiom,
    ! [I: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R )
     => ( member_int @ I @ ( field_int @ R ) ) ) ).

% FieldI1
thf(fact_4516_FieldI1,axiom,
    ! [I: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R: set_Pr2161125870931222855_nat_o] :
      ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I @ J ) @ R )
     => ( member6576561426505652726_nat_o @ I @ ( field_81821745747897882_nat_o @ R ) ) ) ).

% FieldI1
thf(fact_4517_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R )
     => ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R ) ) ) ).

% FieldI1
thf(fact_4518_FieldI1,axiom,
    ! [I: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
     => ( member_nat @ I @ ( field_nat @ R ) ) ) ).

% FieldI1
thf(fact_4519_FieldI1,axiom,
    ! [I: $o,J: $o,R: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R )
     => ( member_o @ I @ ( field_o @ R ) ) ) ).

% FieldI1
thf(fact_4520_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R ) ) ) ).

% FieldI2
thf(fact_4521_FieldI2,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R )
     => ( member8440522571783428010at_nat @ J @ ( field_4706345496283612622at_nat @ R ) ) ) ).

% FieldI2
thf(fact_4522_FieldI2,axiom,
    ! [I: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R )
     => ( member_int @ J @ ( field_int @ R ) ) ) ).

% FieldI2
thf(fact_4523_FieldI2,axiom,
    ! [I: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R: set_Pr2161125870931222855_nat_o] :
      ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I @ J ) @ R )
     => ( member6576561426505652726_nat_o @ J @ ( field_81821745747897882_nat_o @ R ) ) ) ).

% FieldI2
thf(fact_4524_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R ) ) ) ).

% FieldI2
thf(fact_4525_FieldI2,axiom,
    ! [I: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
     => ( member_nat @ J @ ( field_nat @ R ) ) ) ).

% FieldI2
thf(fact_4526_FieldI2,axiom,
    ! [I: $o,J: $o,R: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R )
     => ( member_o @ J @ ( field_o @ R ) ) ) ).

% FieldI2
thf(fact_4527_set__ConsD,axiom,
    ! [Y: produc3843707927480180839at_nat,X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ Y @ ( set_Pr3765526544606949372at_nat @ ( cons_P3940603068885512221at_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member8757157785044589968at_nat @ Y @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4528_set__ConsD,axiom,
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4529_set__ConsD,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( set_Pr400311997697478754_nat_o @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member6576561426505652726_nat_o @ Y @ ( set_Pr400311997697478754_nat_o @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4530_set__ConsD,axiom,
    ! [Y: int,X: int,Xs: list_int] :
      ( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4531_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4532_set__ConsD,axiom,
    ! [Y: produc6575502325842934193n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ Y @ ( set_Pr1139785259514867910n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member7957490590177025114n_assn @ Y @ ( set_Pr1139785259514867910n_assn @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_4533_list_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A2: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat @ E @ ( set_Pr3765526544606949372at_nat @ A2 ) )
     => ( ! [Z22: list_P5464809261938338413at_nat] :
            ( A2
           != ( cons_P3940603068885512221at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc3843707927480180839at_nat,Z22: list_P5464809261938338413at_nat] :
              ( ( A2
                = ( cons_P3940603068885512221at_nat @ Z1 @ Z22 ) )
             => ~ ( member8757157785044589968at_nat @ E @ ( set_Pr3765526544606949372at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4534_list_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A2: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ A2 ) )
     => ( ! [Z22: list_P6011104703257516679at_nat] :
            ( A2
           != ( cons_P6512896166579812791at_nat @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
              ( ( A2
                = ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
             => ~ ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4535_list_Oset__cases,axiom,
    ! [E: produc3658429121746597890et_nat > $o,A2: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ E @ ( set_Pr400311997697478754_nat_o @ A2 ) )
     => ( ! [Z22: list_P7985473006766602707_nat_o] :
            ( A2
           != ( cons_P6219271836124797827_nat_o @ E @ Z22 ) )
       => ~ ! [Z1: produc3658429121746597890et_nat > $o,Z22: list_P7985473006766602707_nat_o] :
              ( ( A2
                = ( cons_P6219271836124797827_nat_o @ Z1 @ Z22 ) )
             => ~ ( member6576561426505652726_nat_o @ E @ ( set_Pr400311997697478754_nat_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4536_list_Oset__cases,axiom,
    ! [E: int,A2: list_int] :
      ( ( member_int @ E @ ( set_int2 @ A2 ) )
     => ( ! [Z22: list_int] :
            ( A2
           != ( cons_int @ E @ Z22 ) )
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A2
                = ( cons_int @ Z1 @ Z22 ) )
             => ~ ( member_int @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4537_list_Oset__cases,axiom,
    ! [E: nat,A2: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A2 ) )
     => ( ! [Z22: list_nat] :
            ( A2
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A2
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4538_list_Oset__cases,axiom,
    ! [E: produc6575502325842934193n_assn,A2: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ E @ ( set_Pr1139785259514867910n_assn @ A2 ) )
     => ( ! [Z22: list_P8527749157015355191n_assn] :
            ( A2
           != ( cons_P2971678138204555879n_assn @ E @ Z22 ) )
       => ~ ! [Z1: produc6575502325842934193n_assn,Z22: list_P8527749157015355191n_assn] :
              ( ( A2
                = ( cons_P2971678138204555879n_assn @ Z1 @ Z22 ) )
             => ~ ( member7957490590177025114n_assn @ E @ ( set_Pr1139785259514867910n_assn @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_4539_list_Oset__intros_I1_J,axiom,
    ! [X21: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat] : ( member8757157785044589968at_nat @ X21 @ ( set_Pr3765526544606949372at_nat @ ( cons_P3940603068885512221at_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4540_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4541_list_Oset__intros_I1_J,axiom,
    ! [X21: produc3658429121746597890et_nat > $o,X222: list_P7985473006766602707_nat_o] : ( member6576561426505652726_nat_o @ X21 @ ( set_Pr400311997697478754_nat_o @ ( cons_P6219271836124797827_nat_o @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4542_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X222: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4543_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4544_list_Oset__intros_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : ( member7957490590177025114n_assn @ X21 @ ( set_Pr1139785259514867910n_assn @ ( cons_P2971678138204555879n_assn @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_4545_list_Oset__intros_I2_J,axiom,
    ! [Y: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat,X21: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ Y @ ( set_Pr3765526544606949372at_nat @ X222 ) )
     => ( member8757157785044589968at_nat @ Y @ ( set_Pr3765526544606949372at_nat @ ( cons_P3940603068885512221at_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4546_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X222: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X222 ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4547_list_Oset__intros_I2_J,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,X222: list_P7985473006766602707_nat_o,X21: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( set_Pr400311997697478754_nat_o @ X222 ) )
     => ( member6576561426505652726_nat_o @ Y @ ( set_Pr400311997697478754_nat_o @ ( cons_P6219271836124797827_nat_o @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4548_list_Oset__intros_I2_J,axiom,
    ! [Y: int,X222: list_int,X21: int] :
      ( ( member_int @ Y @ ( set_int2 @ X222 ) )
     => ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4549_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4550_list_Oset__intros_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn] :
      ( ( member7957490590177025114n_assn @ Y @ ( set_Pr1139785259514867910n_assn @ X222 ) )
     => ( member7957490590177025114n_assn @ Y @ ( set_Pr1139785259514867910n_assn @ ( cons_P2971678138204555879n_assn @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_4551_distinct__length__2__or__more,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( ( distinct_int @ ( cons_int @ A2 @ ( cons_int @ B2 @ Xs ) ) )
      = ( ( A2 != B2 )
        & ( distinct_int @ ( cons_int @ A2 @ Xs ) )
        & ( distinct_int @ ( cons_int @ B2 @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_4552_distinct__length__2__or__more,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ A2 @ ( cons_nat @ B2 @ Xs ) ) )
      = ( ( A2 != B2 )
        & ( distinct_nat @ ( cons_nat @ A2 @ Xs ) )
        & ( distinct_nat @ ( cons_nat @ B2 @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_4553_distinct__length__2__or__more,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( distin4809818325527169450n_assn @ ( cons_P2971678138204555879n_assn @ A2 @ ( cons_P2971678138204555879n_assn @ B2 @ Xs ) ) )
      = ( ( A2 != B2 )
        & ( distin4809818325527169450n_assn @ ( cons_P2971678138204555879n_assn @ A2 @ Xs ) )
        & ( distin4809818325527169450n_assn @ ( cons_P2971678138204555879n_assn @ B2 @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_4554_remove1_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( remove1_int @ X @ ( cons_int @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1_int @ X @ ( cons_int @ Y @ Xs ) )
          = ( cons_int @ Y @ ( remove1_int @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_4555_remove1_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( remove1_nat @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_4556_remove1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( ( remove1670527618125605709n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1670527618125605709n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ Y @ ( remove1670527618125605709n_assn @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_4557_total__onI,axiom,
    ! [A: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A )
         => ( ( member2643936169264416010at_nat @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y2 ) @ R2 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A @ R2 ) ) ).

% total_onI
thf(fact_4558_total__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A )
         => ( ( member8757157785044589968at_nat @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y2 ) @ R2 )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A @ R2 ) ) ).

% total_onI
thf(fact_4559_total__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ( ( member8440522571783428010at_nat @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y2 ) @ R2 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_3592101749530773125at_nat @ A @ R2 ) ) ).

% total_onI
thf(fact_4560_total__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ( member_nat @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_on_nat @ A @ R2 ) ) ).

% total_onI
thf(fact_4561_total__onI,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int] :
          ( ( member_int @ X3 @ A )
         => ( ( member_int @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
                | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_on_int @ A @ R2 ) ) ).

% total_onI
thf(fact_4562_total__onI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A )
         => ( ( member6576561426505652726_nat_o @ Y2 @ A )
           => ( ( X3 != Y2 )
             => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
                | ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
     => ( total_7429123086324524369_nat_o @ A @ R2 ) ) ).

% total_onI
thf(fact_4563_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A5: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A5 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A5 )
             => ( ( X2 != Y5 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R4 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_4564_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A5: set_nat,R4: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A5 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A5 )
             => ( ( X2 != Y5 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R4 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R4 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_4565_total__on__empty,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] : ( total_1513685709984341995at_nat @ bot_bo228742789529271731at_nat @ R2 ) ).

% total_on_empty
thf(fact_4566_total__on__empty,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] : ( total_3592101749530773125at_nat @ bot_bo2099793752762293965at_nat @ R2 ) ).

% total_on_empty
thf(fact_4567_total__on__empty,axiom,
    ! [R2: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R2 ) ).

% total_on_empty
thf(fact_4568_total__on__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R2 ) ).

% total_on_empty
thf(fact_4569_total__on__empty,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R2 ) ).

% total_on_empty
thf(fact_4570_removeAll_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( removeAll_int @ X @ ( cons_int @ Y @ Xs ) )
          = ( removeAll_int @ X @ Xs ) ) )
      & ( ( X != Y )
       => ( ( removeAll_int @ X @ ( cons_int @ Y @ Xs ) )
          = ( cons_int @ Y @ ( removeAll_int @ X @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_4571_removeAll_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = ( removeAll_nat @ X @ Xs ) ) )
      & ( ( X != Y )
       => ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( removeAll_nat @ X @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_4572_removeAll_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( ( remove1763081955827946941n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = ( remove1763081955827946941n_assn @ X @ Xs ) ) )
      & ( ( X != Y )
       => ( ( remove1763081955827946941n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ Y @ ( remove1763081955827946941n_assn @ X @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_4573_zipf_Osimps_I2_J,axiom,
    ! [F4: int > int > int,A2: int,As2: list_int,B2: int,Bs: list_int] :
      ( ( zipf_int_int_int @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_int @ B2 @ Bs ) )
      = ( cons_int @ ( F4 @ A2 @ B2 ) @ ( zipf_int_int_int @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4574_zipf_Osimps_I2_J,axiom,
    ! [F4: int > int > nat,A2: int,As2: list_int,B2: int,Bs: list_int] :
      ( ( zipf_int_int_nat @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_int @ B2 @ Bs ) )
      = ( cons_nat @ ( F4 @ A2 @ B2 ) @ ( zipf_int_int_nat @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4575_zipf_Osimps_I2_J,axiom,
    ! [F4: int > nat > int,A2: int,As2: list_int,B2: nat,Bs: list_nat] :
      ( ( zipf_int_nat_int @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_nat @ B2 @ Bs ) )
      = ( cons_int @ ( F4 @ A2 @ B2 ) @ ( zipf_int_nat_int @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4576_zipf_Osimps_I2_J,axiom,
    ! [F4: int > nat > nat,A2: int,As2: list_int,B2: nat,Bs: list_nat] :
      ( ( zipf_int_nat_nat @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_nat @ B2 @ Bs ) )
      = ( cons_nat @ ( F4 @ A2 @ B2 ) @ ( zipf_int_nat_nat @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4577_zipf_Osimps_I2_J,axiom,
    ! [F4: nat > int > int,A2: nat,As2: list_nat,B2: int,Bs: list_int] :
      ( ( zipf_nat_int_int @ F4 @ ( cons_nat @ A2 @ As2 ) @ ( cons_int @ B2 @ Bs ) )
      = ( cons_int @ ( F4 @ A2 @ B2 ) @ ( zipf_nat_int_int @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4578_zipf_Osimps_I2_J,axiom,
    ! [F4: nat > int > nat,A2: nat,As2: list_nat,B2: int,Bs: list_int] :
      ( ( zipf_nat_int_nat @ F4 @ ( cons_nat @ A2 @ As2 ) @ ( cons_int @ B2 @ Bs ) )
      = ( cons_nat @ ( F4 @ A2 @ B2 ) @ ( zipf_nat_int_nat @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4579_zipf_Osimps_I2_J,axiom,
    ! [F4: nat > nat > int,A2: nat,As2: list_nat,B2: nat,Bs: list_nat] :
      ( ( zipf_nat_nat_int @ F4 @ ( cons_nat @ A2 @ As2 ) @ ( cons_nat @ B2 @ Bs ) )
      = ( cons_int @ ( F4 @ A2 @ B2 ) @ ( zipf_nat_nat_int @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4580_zipf_Osimps_I2_J,axiom,
    ! [F4: nat > nat > nat,A2: nat,As2: list_nat,B2: nat,Bs: list_nat] :
      ( ( zipf_nat_nat_nat @ F4 @ ( cons_nat @ A2 @ As2 ) @ ( cons_nat @ B2 @ Bs ) )
      = ( cons_nat @ ( F4 @ A2 @ B2 ) @ ( zipf_nat_nat_nat @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4581_zipf_Osimps_I2_J,axiom,
    ! [F4: int > int > produc6575502325842934193n_assn,A2: int,As2: list_int,B2: int,Bs: list_int] :
      ( ( zipf_i7756021159461276697n_assn @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_int @ B2 @ Bs ) )
      = ( cons_P2971678138204555879n_assn @ ( F4 @ A2 @ B2 ) @ ( zipf_i7756021159461276697n_assn @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4582_zipf_Osimps_I2_J,axiom,
    ! [F4: int > nat > produc6575502325842934193n_assn,A2: int,As2: list_int,B2: nat,Bs: list_nat] :
      ( ( zipf_i5995984495708165493n_assn @ F4 @ ( cons_int @ A2 @ As2 ) @ ( cons_nat @ B2 @ Bs ) )
      = ( cons_P2971678138204555879n_assn @ ( F4 @ A2 @ B2 ) @ ( zipf_i5995984495708165493n_assn @ F4 @ As2 @ Bs ) ) ) ).

% zipf.simps(2)
thf(fact_4583_null__rec_I1_J,axiom,
    ! [X: int,Xs: list_int] :
      ~ ( null_int @ ( cons_int @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_4584_null__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_4585_null__rec_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ~ ( null_P7596217980778516671n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_4586_length__Suc__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y5 @ Ys3 ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4587_length__Suc__conv,axiom,
    ! [Xs: list_c,N: nat] :
      ( ( ( size_size_list_c @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: c,Ys3: list_c] :
            ( ( Xs
              = ( cons_c @ Y5 @ Ys3 ) )
            & ( ( size_size_list_c @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4588_length__Suc__conv,axiom,
    ! [Xs: list_d,N: nat] :
      ( ( ( size_size_list_d @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: d,Ys3: list_d] :
            ( ( Xs
              = ( cons_d @ Y5 @ Ys3 ) )
            & ( ( size_size_list_d @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4589_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4590_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y5 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_4591_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ( suc @ N )
        = ( size_s6829681357464350627n_assn @ Xs ) )
      = ( ? [Y5: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y5 @ Ys3 ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4592_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( ( suc @ N )
        = ( size_size_list_c @ Xs ) )
      = ( ? [Y5: c,Ys3: list_c] :
            ( ( Xs
              = ( cons_c @ Y5 @ Ys3 ) )
            & ( ( size_size_list_c @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4593_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( ( suc @ N )
        = ( size_size_list_d @ Xs ) )
      = ( ? [Y5: d,Ys3: list_d] :
            ( ( Xs
              = ( cons_d @ Y5 @ Ys3 ) )
            & ( ( size_size_list_d @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4594_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4595_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs ) )
      = ( ? [Y5: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y5 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_4596_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_4597_set__subset__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] : ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ ( set_Pr1139785259514867910n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_4598_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_4599_mono__Field,axiom,
    ! [R2: set_Product_prod_o_o,S3: set_Product_prod_o_o] :
      ( ( ord_le192195778054999463od_o_o @ R2 @ S3 )
     => ( ord_less_eq_set_o @ ( field_o @ R2 ) @ ( field_o @ S3 ) ) ) ).

% mono_Field
thf(fact_4600_mono__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
     => ( ord_less_eq_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S3 ) ) ) ).

% mono_Field
thf(fact_4601_impossible__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ord_less_eq_nat @ ( size_s6829681357464350627n_assn @ Xs ) @ ( size_s6829681357464350627n_assn @ Ys ) )
     => ( Xs
       != ( cons_P2971678138204555879n_assn @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4602_impossible__Cons,axiom,
    ! [Xs: list_c,Ys: list_c,X: c] :
      ( ( ord_less_eq_nat @ ( size_size_list_c @ Xs ) @ ( size_size_list_c @ Ys ) )
     => ( Xs
       != ( cons_c @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4603_impossible__Cons,axiom,
    ! [Xs: list_d,Ys: list_d,X: d] :
      ( ( ord_less_eq_nat @ ( size_size_list_d @ Xs ) @ ( size_size_list_d @ Ys ) )
     => ( Xs
       != ( cons_d @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4604_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4605_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
     => ( Xs
       != ( cons_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_4606_sorted2,axiom,
    ! [X: nat,Y: nat,Zs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4607_sorted2,axiom,
    ! [X: int,Y: int,Zs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ ( cons_int @ Y @ Zs ) ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_4608_distinct_Osimps_I2_J,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( distin5355857245021852384at_nat @ ( cons_P3940603068885512221at_nat @ X @ Xs ) )
      = ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
        & ( distin5355857245021852384at_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4609_distinct_Osimps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
      = ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( distin6923225563576452346at_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4610_distinct_Osimps_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ( distin299582740263731014_nat_o @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) )
      = ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
        & ( distin299582740263731014_nat_o @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4611_distinct_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( distinct_int @ ( cons_int @ X @ Xs ) )
      = ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
        & ( distinct_int @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4612_distinct_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
      = ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4613_distinct_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( distin4809818325527169450n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
        & ( distin4809818325527169450n_assn @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_4614_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_int @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_4615_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_4616_list__update__code_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ( list_u4534839942911652127n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_P2971678138204555879n_assn @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_4617_list__update__code_I2_J,axiom,
    ! [X: c,Xs: list_c,Y: c] :
      ( ( list_update_c @ ( cons_c @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_c @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_4618_list__update__code_I2_J,axiom,
    ! [X: d,Xs: list_d,Y: d] :
      ( ( list_update_d @ ( cons_d @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_d @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_4619_list__update__code_I3_J,axiom,
    ! [X: int,Xs: list_int,I: nat,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_int @ X @ ( list_update_int @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4620_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4621_list__update__code_I3_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] :
      ( ( list_u4534839942911652127n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_P2971678138204555879n_assn @ X @ ( list_u4534839942911652127n_assn @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4622_list__update__code_I3_J,axiom,
    ! [X: c,Xs: list_c,I: nat,Y: c] :
      ( ( list_update_c @ ( cons_c @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_c @ X @ ( list_update_c @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4623_list__update__code_I3_J,axiom,
    ! [X: d,Xs: list_d,I: nat,Y: d] :
      ( ( list_update_d @ ( cons_d @ X @ Xs ) @ ( suc @ I ) @ Y )
      = ( cons_d @ X @ ( list_update_d @ Xs @ I @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4624_listrel1I2,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,X: int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ X @ Ys ) ) @ ( listrel1_int @ R2 ) ) ) ).

% listrel1I2
thf(fact_4625_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% listrel1I2
thf(fact_4626_listrel1I2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn,X: produc6575502325842934193n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ Ys ) @ ( listre4270255209692100112n_assn @ R2 ) )
     => ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ X @ Ys ) ) @ ( listre4270255209692100112n_assn @ R2 ) ) ) ).

% listrel1I2
thf(fact_4627_merge_Osimps_I3_J,axiom,
    ! [X1: code_integer,X22: code_integer,L12: list_Code_integer,L23: list_Code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ X1 @ X22 )
       => ( ( merge_Code_integer @ ( cons_Code_integer @ X1 @ L12 ) @ ( cons_Code_integer @ X22 @ L23 ) )
          = ( cons_Code_integer @ X1 @ ( merge_Code_integer @ L12 @ ( cons_Code_integer @ X22 @ L23 ) ) ) ) )
      & ( ~ ( ord_le6747313008572928689nteger @ X1 @ X22 )
       => ( ( ( X1 = X22 )
           => ( ( merge_Code_integer @ ( cons_Code_integer @ X1 @ L12 ) @ ( cons_Code_integer @ X22 @ L23 ) )
              = ( cons_Code_integer @ X1 @ ( merge_Code_integer @ L12 @ L23 ) ) ) )
          & ( ( X1 != X22 )
           => ( ( merge_Code_integer @ ( cons_Code_integer @ X1 @ L12 ) @ ( cons_Code_integer @ X22 @ L23 ) )
              = ( cons_Code_integer @ X22 @ ( merge_Code_integer @ ( cons_Code_integer @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_4628_merge_Osimps_I3_J,axiom,
    ! [X1: num,X22: num,L12: list_num,L23: list_num] :
      ( ( ( ord_less_num @ X1 @ X22 )
       => ( ( merge_num @ ( cons_num @ X1 @ L12 ) @ ( cons_num @ X22 @ L23 ) )
          = ( cons_num @ X1 @ ( merge_num @ L12 @ ( cons_num @ X22 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_num @ X1 @ X22 )
       => ( ( ( X1 = X22 )
           => ( ( merge_num @ ( cons_num @ X1 @ L12 ) @ ( cons_num @ X22 @ L23 ) )
              = ( cons_num @ X1 @ ( merge_num @ L12 @ L23 ) ) ) )
          & ( ( X1 != X22 )
           => ( ( merge_num @ ( cons_num @ X1 @ L12 ) @ ( cons_num @ X22 @ L23 ) )
              = ( cons_num @ X22 @ ( merge_num @ ( cons_num @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_4629_merge_Osimps_I3_J,axiom,
    ! [X1: nat,X22: nat,L12: list_nat,L23: list_nat] :
      ( ( ( ord_less_nat @ X1 @ X22 )
       => ( ( merge_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X22 @ L23 ) )
          = ( cons_nat @ X1 @ ( merge_nat @ L12 @ ( cons_nat @ X22 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_nat @ X1 @ X22 )
       => ( ( ( X1 = X22 )
           => ( ( merge_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X22 @ L23 ) )
              = ( cons_nat @ X1 @ ( merge_nat @ L12 @ L23 ) ) ) )
          & ( ( X1 != X22 )
           => ( ( merge_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X22 @ L23 ) )
              = ( cons_nat @ X22 @ ( merge_nat @ ( cons_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_4630_merge_Osimps_I3_J,axiom,
    ! [X1: int,X22: int,L12: list_int,L23: list_int] :
      ( ( ( ord_less_int @ X1 @ X22 )
       => ( ( merge_int @ ( cons_int @ X1 @ L12 ) @ ( cons_int @ X22 @ L23 ) )
          = ( cons_int @ X1 @ ( merge_int @ L12 @ ( cons_int @ X22 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_int @ X1 @ X22 )
       => ( ( ( X1 = X22 )
           => ( ( merge_int @ ( cons_int @ X1 @ L12 ) @ ( cons_int @ X22 @ L23 ) )
              = ( cons_int @ X1 @ ( merge_int @ L12 @ L23 ) ) ) )
          & ( ( X1 != X22 )
           => ( ( merge_int @ ( cons_int @ X1 @ L12 ) @ ( cons_int @ X22 @ L23 ) )
              = ( cons_int @ X22 @ ( merge_int @ ( cons_int @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_4631_list__assn_Osimps_I2_J,axiom,
    ! [P: int > int > assn,A2: int,As2: list_int,C: int,Cs: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L74593716426352029nt_int @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4632_list__assn_Osimps_I2_J,axiom,
    ! [P: int > nat > assn,A2: int,As2: list_int,C: nat,Cs: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L77084186935402305nt_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4633_list__assn_Osimps_I2_J,axiom,
    ! [P: int > produc6575502325842934193n_assn > assn,A2: int,As2: list_int,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L1620274777744359680n_assn @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L1620274777744359680n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4634_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > int > assn,A2: nat,As2: list_nat,C: int,Cs: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L8298612041380073281at_int @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4635_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > nat > assn,A2: nat,As2: list_nat,C: nat,Cs: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L8301102511889123557at_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4636_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > produc6575502325842934193n_assn > assn,A2: nat,As2: list_nat,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L9083610150846024284n_assn @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L9083610150846024284n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4637_list__assn_Osimps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > int > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: int,Cs: list_int] :
      ( ( vEBT_L4760345170377931482sn_int @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L4760345170377931482sn_int @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4638_list__assn_Osimps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > nat > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: nat,Cs: list_nat] :
      ( ( vEBT_L4762835640886981758sn_nat @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L4762835640886981758sn_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4639_list__assn_Osimps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L3108143461992071043n_assn @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L3108143461992071043n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_4640_list__assn__simps_I2_J,axiom,
    ! [P: int > int > assn,A2: int,As2: list_int,C: int,Cs: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L74593716426352029nt_int @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4641_list__assn__simps_I2_J,axiom,
    ! [P: int > nat > assn,A2: int,As2: list_int,C: nat,Cs: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L77084186935402305nt_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4642_list__assn__simps_I2_J,axiom,
    ! [P: int > produc6575502325842934193n_assn > assn,A2: int,As2: list_int,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L1620274777744359680n_assn @ P @ ( cons_int @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L1620274777744359680n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4643_list__assn__simps_I2_J,axiom,
    ! [P: nat > int > assn,A2: nat,As2: list_nat,C: int,Cs: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L8298612041380073281at_int @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4644_list__assn__simps_I2_J,axiom,
    ! [P: nat > nat > assn,A2: nat,As2: list_nat,C: nat,Cs: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L8301102511889123557at_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4645_list__assn__simps_I2_J,axiom,
    ! [P: nat > produc6575502325842934193n_assn > assn,A2: nat,As2: list_nat,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L9083610150846024284n_assn @ P @ ( cons_nat @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L9083610150846024284n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4646_list__assn__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > int > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: int,Cs: list_int] :
      ( ( vEBT_L4760345170377931482sn_int @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_int @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L4760345170377931482sn_int @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4647_list__assn__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > nat > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: nat,Cs: list_nat] :
      ( ( vEBT_L4762835640886981758sn_nat @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_nat @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L4762835640886981758sn_nat @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4648_list__assn__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] :
      ( ( vEBT_L3108143461992071043n_assn @ P @ ( cons_P2971678138204555879n_assn @ A2 @ As2 ) @ ( cons_P2971678138204555879n_assn @ C @ Cs ) )
      = ( times_times_assn @ ( P @ A2 @ C ) @ ( vEBT_L3108143461992071043n_assn @ P @ As2 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_4649_List_Oinsert__def,axiom,
    ( insert4149901806602606979at_nat
    = ( ^ [X2: produc3843707927480180839at_nat,Xs3: list_P5464809261938338413at_nat] : ( if_lis7508161100780771379at_nat @ ( member8757157785044589968at_nat @ X2 @ ( set_Pr3765526544606949372at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P3940603068885512221at_nat @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4650_List_Oinsert__def,axiom,
    ( insert8944034826898310173at_nat
    = ( ^ [X2: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P6512896166579812791at_nat @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4651_List_Oinsert__def,axiom,
    ( insert8021100735160035049_nat_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o,Xs3: list_P7985473006766602707_nat_o] : ( if_lis2904118647880296857_nat_o @ ( member6576561426505652726_nat_o @ X2 @ ( set_Pr400311997697478754_nat_o @ Xs3 ) ) @ Xs3 @ ( cons_P6219271836124797827_nat_o @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4652_List_Oinsert__def,axiom,
    ( insert_int
    = ( ^ [X2: int,Xs3: list_int] : ( if_list_int @ ( member_int @ X2 @ ( set_int2 @ Xs3 ) ) @ Xs3 @ ( cons_int @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4653_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X2: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X2 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4654_List_Oinsert__def,axiom,
    ( insert3246601298802261197n_assn
    = ( ^ [X2: produc6575502325842934193n_assn,Xs3: list_P8527749157015355191n_assn] : ( if_lis7173127979115136509n_assn @ ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs3 ) ) @ Xs3 @ ( cons_P2971678138204555879n_assn @ X2 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_4655_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6829681357464350627n_assn @ Xs ) )
      = ( ? [X2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6829681357464350627n_assn @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4656_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_c @ Xs ) )
      = ( ? [X2: c,Ys3: list_c] :
            ( ( Xs
              = ( cons_c @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_c @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4657_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_d @ Xs ) )
      = ( ? [X2: d,Ys3: list_d] :
            ( ( Xs
              = ( cons_d @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_d @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4658_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X2: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4659_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X2: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X2 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_4660_sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ( ord_less_eq_nat @ X @ X2 ) )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4661_sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ Ys ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Ys ) )
           => ( ord_less_eq_int @ X @ X2 ) )
        & ( sorted_wrt_int @ ord_less_eq_int @ Ys ) ) ) ).

% sorted_simps(2)
thf(fact_4662_strict__sorted__simps_I2_J,axiom,
    ! [X: code_integer,Ys: list_Code_integer] :
      ( ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ ( cons_Code_integer @ X @ Ys ) )
      = ( ! [X2: code_integer] :
            ( ( member_Code_integer @ X2 @ ( set_Code_integer2 @ Ys ) )
           => ( ord_le6747313008572928689nteger @ X @ X2 ) )
        & ( sorted710888440204495920nteger @ ord_le6747313008572928689nteger @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4663_strict__sorted__simps_I2_J,axiom,
    ! [X: num,Ys: list_num] :
      ( ( sorted_wrt_num @ ord_less_num @ ( cons_num @ X @ Ys ) )
      = ( ! [X2: num] :
            ( ( member_num @ X2 @ ( set_num2 @ Ys ) )
           => ( ord_less_num @ X @ X2 ) )
        & ( sorted_wrt_num @ ord_less_num @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4664_strict__sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ( ord_less_nat @ X @ X2 ) )
        & ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4665_strict__sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ ( cons_int @ X @ Ys ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Ys ) )
           => ( ord_less_int @ X @ X2 ) )
        & ( sorted_wrt_int @ ord_less_int @ Ys ) ) ) ).

% strict_sorted_simps(2)
thf(fact_4666_bijective__def,axiom,
    ( biject3388518990857851981at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
          ( ! [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat,Z5: set_Pr1261947904930325089at_nat] :
              ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R3 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat,Z5: set_Pr1261947904930325089at_nat] :
              ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z5 ) @ R3 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4667_bijective__def,axiom,
    ( bijective_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( ! [X2: nat,Y5: nat,Z5: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: nat,Y5: nat,Z5: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z5 ) @ R3 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4668_bijective__def,axiom,
    ( biject9051520373387432658nteger
    = ( ^ [R3: set_Pr1281608226676607948nteger] :
          ( ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger,Z5: produc8923325533196201883nteger] :
              ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y5 ) @ R3 )
                & ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc6241069584506657477e_term > option6357759511663192854e_term,Z5: produc8923325533196201883nteger] :
              ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Z5 ) @ R3 )
                & ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4669_bijective__def,axiom,
    ( biject2615096655818420098et_nat
    = ( ^ [R3: set_Pr3286484037609594932et_nat] :
          ( ! [X2: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat,Z5: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y5 ) @ R3 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat > $o,Z5: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z5 ) @ R3 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4670_bijective__def,axiom,
    ( biject1468766312547416318et_nat
    = ( ^ [R3: set_Pr8536935166611901872et_nat] :
          ( ! [X2: produc3658429121746597890et_nat > $o,Y5: produc3925858234332021118et_nat,Z5: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y5 ) @ R3 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat > $o,Z5: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z5 ) @ R3 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4671_bijective__def,axiom,
    ( biject383251550997737151nt_int
    = ( ^ [R3: set_Pr9222295170931077689nt_int] :
          ( ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int,Z5: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y5 ) @ R3 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: produc8551481072490612790e_term > option6357759511663192854e_term,Z5: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Z5 ) @ R3 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4672_bijective__def,axiom,
    ( biject576505603616484041nt_int
    = ( ^ [R3: set_Pr1872883991513573699nt_int] :
          ( ! [X2: int > option6357759511663192854e_term,Y5: product_prod_int_int,Z5: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y5 ) @ R3 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Z5 ) @ R3 ) )
             => ( Y5 = Z5 ) )
          & ! [X2: int > option6357759511663192854e_term,Y5: int > option6357759511663192854e_term,Z5: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Z5 ) @ R3 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y5 @ Z5 ) @ R3 ) )
             => ( X2 = Y5 ) ) ) ) ) ).

% bijective_def
thf(fact_4673_Cons__listrel1E2,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre8180072825634266816at_nat @ R2 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( Xs
              = ( cons_s6881495754146722583at_nat @ X3 @ Ys ) )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y ) @ R2 ) )
       => ~ ! [Zs2: list_s1210847774152347623at_nat] :
              ( ( Xs
                = ( cons_s6881495754146722583at_nat @ Y @ Zs2 ) )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Zs2 @ Ys ) @ ( listre8180072825634266816at_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_4674_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R2 ) )
     => ( ! [X3: int] :
            ( ( Xs
              = ( cons_int @ X3 @ Ys ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R2 ) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs
                = ( cons_int @ Y @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_4675_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
     => ( ! [X3: nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_4676_Cons__listrel1E2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre4270255209692100112n_assn @ R2 ) )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Ys ) )
           => ~ ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X3 @ Y ) @ R2 ) )
       => ~ ! [Zs2: list_P8527749157015355191n_assn] :
              ( ( Xs
                = ( cons_P2971678138204555879n_assn @ Y @ Zs2 ) )
             => ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Zs2 @ Ys ) @ ( listre4270255209692100112n_assn @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_4677_Cons__listrel1E1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ Ys ) @ ( listre8180072825634266816at_nat @ R2 ) )
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( ( Ys
              = ( cons_s6881495754146722583at_nat @ Y2 @ Xs ) )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y2 ) @ R2 ) )
       => ~ ! [Zs2: list_s1210847774152347623at_nat] :
              ( ( Ys
                = ( cons_s6881495754146722583at_nat @ X @ Zs2 ) )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Zs2 ) @ ( listre8180072825634266816at_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_4678_Cons__listrel1E1,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) @ ( listrel1_int @ R2 ) )
     => ( ! [Y2: int] :
            ( ( Ys
              = ( cons_int @ Y2 @ Xs ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 ) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys
                = ( cons_int @ X @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_4679_Cons__listrel1E1,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ! [Y2: nat] :
            ( ( Ys
              = ( cons_nat @ Y2 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_4680_Cons__listrel1E1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Ys ) @ ( listre4270255209692100112n_assn @ R2 ) )
     => ( ! [Y2: produc6575502325842934193n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Y2 @ Xs ) )
           => ~ ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X @ Y2 ) @ R2 ) )
       => ~ ! [Zs2: list_P8527749157015355191n_assn] :
              ( ( Ys
                = ( cons_P2971678138204555879n_assn @ X @ Zs2 ) )
             => ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ Zs2 ) @ ( listre4270255209692100112n_assn @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_4681_listrel1I1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ ( cons_s6881495754146722583at_nat @ Y @ Xs ) ) @ ( listre8180072825634266816at_nat @ R2 ) ) ) ).

% listrel1I1
thf(fact_4682_listrel1I1,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R2 ) ) ) ).

% listrel1I1
thf(fact_4683_listrel1I1,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% listrel1I1
thf(fact_4684_listrel1I1,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,R2: set_Pr8116311503512020807n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X @ Y ) @ R2 )
     => ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) ) @ ( listre4270255209692100112n_assn @ R2 ) ) ) ).

% listrel1I1
thf(fact_4685_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R2 ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R2 )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4686_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: int,Ys: list_int,R2: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel_nat_int @ R2 ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ Y ) @ R2 )
             => ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs2 @ Ys ) @ ( listrel_nat_int @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4687_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: nat,Ys: list_nat,R2: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_int_nat @ R2 ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ Y ) @ R2 )
             => ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs2 @ Ys ) @ ( listrel_int_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4688_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4689_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int,R2: set_Pr2181989527757714242sn_int] :
      ( ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listre8029020076619294380sn_int @ R2 ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) )
           => ( ( member5792320502188913699sn_int @ ( produc6453891449406693510sn_int @ X3 @ Y ) @ R2 )
             => ~ ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ Xs2 @ Ys ) @ ( listre8029020076619294380sn_int @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4690_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat,R2: set_Pr4672072672689440998sn_nat] :
      ( ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listre8031510547128344656sn_nat @ R2 ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) )
           => ( ( member746799484843334599sn_nat @ ( produc6456381919915743786sn_nat @ X3 @ Y ) @ R2 )
             => ~ ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ Xs2 @ Ys ) @ ( listre8031510547128344656sn_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4691_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr3255386646789011632n_assn] :
      ( ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre4888949683985722578n_assn @ R2 ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ( ( member241480614033477777n_assn @ ( produc3313821056773121708n_assn @ X3 @ Y ) @ R2 )
             => ~ ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ Xs2 @ Ys ) @ ( listre4888949683985722578n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4692_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr247834672394327052n_assn] :
      ( ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre3128913020232611374n_assn @ R2 ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member7704815987135142381n_assn @ ( produc1553784393020010504n_assn @ X3 @ Y ) @ R2 )
             => ~ ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ Xs2 @ Ys ) @ ( listre3128913020232611374n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4693_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre5253995379918703921n_assn @ R2 ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) )
           => ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X3 @ Y ) @ R2 )
             => ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs2 @ Ys ) @ ( listre5253995379918703921n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4694_listrel__Cons2,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre2047417242196832561at_nat @ R2 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
            ( ( Xs
              = ( cons_s6881495754146722583at_nat @ X3 @ Xs2 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y ) @ R2 )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs2 @ Ys ) @ ( listre2047417242196832561at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_4695_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listrel_int_int @ R2 ) )
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs
              = ( cons_int @ Y2 @ Ys5 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y2 ) @ R2 )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys5 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4696_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_nat,R2: set_Pr3448869479623346877nt_nat] :
      ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listrel_int_nat @ R2 ) )
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys5 ) )
           => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ Y @ Y2 ) @ R2 )
             => ~ ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Ys @ Ys5 ) @ ( listrel_int_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4697_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_int,R2: set_Pr7995236796853374141at_int] :
      ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_int @ R2 ) )
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs
              = ( cons_int @ Y2 @ Ys5 ) )
           => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ Y @ Y2 ) @ R2 )
             => ~ ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Ys @ Ys5 ) @ ( listrel_nat_int @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4698_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys5 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y2 ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4699_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_P8527749157015355191n_assn,R2: set_Pr3255386646789011632n_assn] :
      ( ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listre4888949683985722578n_assn @ R2 ) )
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
           => ( ( member241480614033477777n_assn @ ( produc3313821056773121708n_assn @ Y @ Y2 ) @ R2 )
             => ~ ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ Ys @ Ys5 ) @ ( listre4888949683985722578n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4700_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_P8527749157015355191n_assn,R2: set_Pr247834672394327052n_assn] :
      ( ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listre3128913020232611374n_assn @ R2 ) )
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
           => ( ( member7704815987135142381n_assn @ ( produc1553784393020010504n_assn @ Y @ Y2 ) @ R2 )
             => ~ ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ Ys @ Ys5 ) @ ( listre3128913020232611374n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4701_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_int,R2: set_Pr2181989527757714242sn_int] :
      ( ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) @ Xs ) @ ( listre8029020076619294380sn_int @ R2 ) )
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs
              = ( cons_int @ Y2 @ Ys5 ) )
           => ( ( member5792320502188913699sn_int @ ( produc6453891449406693510sn_int @ Y @ Y2 ) @ R2 )
             => ~ ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ Ys @ Ys5 ) @ ( listre8029020076619294380sn_int @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4702_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_nat,R2: set_Pr4672072672689440998sn_nat] :
      ( ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) @ Xs ) @ ( listre8031510547128344656sn_nat @ R2 ) )
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y2 @ Ys5 ) )
           => ( ( member746799484843334599sn_nat @ ( produc6456381919915743786sn_nat @ Y @ Y2 ) @ R2 )
             => ~ ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ Ys @ Ys5 ) @ ( listre8031510547128344656sn_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4703_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) @ Xs ) @ ( listre5253995379918703921n_assn @ R2 ) )
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
           => ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ Y @ Y2 ) @ R2 )
             => ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Ys @ Ys5 ) @ ( listre5253995379918703921n_assn @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4704_listrel__Cons1,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Xs: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) @ Xs ) @ ( listre2047417242196832561at_nat @ R2 ) )
     => ~ ! [Y2: set_Pr1261947904930325089at_nat,Ys5: list_s1210847774152347623at_nat] :
            ( ( Xs
              = ( cons_s6881495754146722583at_nat @ Y2 @ Ys5 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Y2 ) @ R2 )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Ys @ Ys5 ) @ ( listre2047417242196832561at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_4705_listrel_OCons,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R2 ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4706_listrel_OCons,axiom,
    ! [X: int,Y: nat,R2: set_Pr3448869479623346877nt_nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ R2 )
     => ( ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ Xs @ Ys ) @ ( listrel_int_nat @ R2 ) )
       => ( member9189046780804443046st_nat @ ( produc4542114716404682293st_nat @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_int_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4707_listrel_OCons,axiom,
    ! [X: nat,Y: int,R2: set_Pr7995236796853374141at_int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y ) @ R2 )
     => ( ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ Xs @ Ys ) @ ( listrel_nat_int @ R2 ) )
       => ( member4850886304473975718st_int @ ( produc7739558402351520821st_int @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_nat_int @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4708_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4709_listrel_OCons,axiom,
    ! [X: int,Y: produc6575502325842934193n_assn,R2: set_Pr3255386646789011632n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( member241480614033477777n_assn @ ( produc3313821056773121708n_assn @ X @ Y ) @ R2 )
     => ( ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ Xs @ Ys ) @ ( listre4888949683985722578n_assn @ R2 ) )
       => ( member490958779117458567n_assn @ ( produc4510294837168919970n_assn @ ( cons_int @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre4888949683985722578n_assn @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4710_listrel_OCons,axiom,
    ! [X: nat,Y: produc6575502325842934193n_assn,R2: set_Pr247834672394327052n_assn,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( member7704815987135142381n_assn @ ( produc1553784393020010504n_assn @ X @ Y ) @ R2 )
     => ( ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ Xs @ Ys ) @ ( listre3128913020232611374n_assn @ R2 ) )
       => ( member3687432546746012131n_assn @ ( produc7706768604797473534n_assn @ ( cons_nat @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre3128913020232611374n_assn @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4711_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: int,R2: set_Pr2181989527757714242sn_int,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( member5792320502188913699sn_int @ ( produc6453891449406693510sn_int @ X @ Y ) @ R2 )
     => ( ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ Xs @ Ys ) @ ( listre8029020076619294380sn_int @ R2 ) )
       => ( member5295696281703116473st_int @ ( produc1996262876970182428st_int @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listre8029020076619294380sn_int @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4712_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: nat,R2: set_Pr4672072672689440998sn_nat,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( member746799484843334599sn_nat @ ( produc6456381919915743786sn_nat @ X @ Y ) @ R2 )
     => ( ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ Xs @ Ys ) @ ( listre8031510547128344656sn_nat @ R2 ) )
       => ( member7785779426634843229st_nat @ ( produc6174113896479379136st_nat @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listre8031510547128344656sn_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4713_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,R2: set_Pr8116311503512020807n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X @ Y ) @ R2 )
     => ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ Ys ) @ ( listre5253995379918703921n_assn @ R2 ) )
       => ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( listre5253995379918703921n_assn @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4714_listrel_OCons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( listre2047417242196832561at_nat @ R2 ) )
       => ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre2047417242196832561at_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_4715_list_Osize_I4_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn @ ( cons_P2971678138204555879n_assn @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_s6829681357464350627n_assn @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_4716_list_Osize_I4_J,axiom,
    ! [X21: c,X222: list_c] :
      ( ( size_size_list_c @ ( cons_c @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_c @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_4717_list_Osize_I4_J,axiom,
    ! [X21: d,X222: list_d] :
      ( ( size_size_list_d @ ( cons_d @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_d @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_4718_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_4719_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_4720_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4721_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4722_nth__Cons_H,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_4723_nth__equal__first__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat,N: nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s1588839187904960729at_nat @ Xs ) )
       => ( ( ( nth_Pr1239524528086697198at_nat @ ( cons_P3940603068885512221at_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4724_nth__equal__first__eq,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
       => ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4725_nth__equal__first__eq,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,N: nat] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s8908918004648414271_nat_o @ Xs ) )
       => ( ( ( nth_Pr6840583978799713108_nat_o @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4726_nth__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
       => ( ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4727_nth__equal__first__eq,axiom,
    ! [X: c,Xs: list_c,N: nat] :
      ( ~ ( member_c @ X @ ( set_c2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_c @ Xs ) )
       => ( ( ( nth_c @ ( cons_c @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4728_nth__equal__first__eq,axiom,
    ! [X: d,Xs: list_d,N: nat] :
      ( ~ ( member_d @ X @ ( set_d2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_d @ Xs ) )
       => ( ( ( nth_d @ ( cons_d @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4729_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4730_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_4731_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4732_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4733_nth__non__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_4734_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_4735_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_4736_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_7964966981652229398n_assn @ Begin @ End @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ X @ ( slice_7964966981652229398n_assn @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_7964966981652229398n_assn @ Begin @ End @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( slice_7964966981652229398n_assn @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_4737_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As22: set_nat] :
                        ( ( As
                          = ( sup_sup_set_nat @ As1 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_4738_length__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( suc @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ).

% length_Cons
thf(fact_4739_length__Cons,axiom,
    ! [X: c,Xs: list_c] :
      ( ( size_size_list_c @ ( cons_c @ X @ Xs ) )
      = ( suc @ ( size_size_list_c @ Xs ) ) ) ).

% length_Cons
thf(fact_4740_length__Cons,axiom,
    ! [X: d,Xs: list_d] :
      ( ( size_size_list_d @ ( cons_d @ X @ Xs ) )
      = ( suc @ ( size_size_list_d @ Xs ) ) ) ).

% length_Cons
thf(fact_4741_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_4742_length__Cons,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_Cons
thf(fact_4743_Cons__lenlex__iff,axiom,
    ! [M2: set_Pr1261947904930325089at_nat,Ms: list_s1210847774152347623at_nat,N: set_Pr1261947904930325089at_nat,Ns: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ M2 @ Ms ) @ ( cons_s6881495754146722583at_nat @ N @ Ns ) ) @ ( lenlex1357538814655152620at_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ms ) @ ( size_s8736152011456118867at_nat @ Ns ) )
        | ( ( ( size_s8736152011456118867at_nat @ Ms )
            = ( size_s8736152011456118867at_nat @ Ns ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Ms @ Ns ) @ ( lenlex1357538814655152620at_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4744_Cons__lenlex__iff,axiom,
    ! [M2: produc6575502325842934193n_assn,Ms: list_P8527749157015355191n_assn,N: produc6575502325842934193n_assn,Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ M2 @ Ms ) @ ( cons_P2971678138204555879n_assn @ N @ Ns ) ) @ ( lenlex6671093235567761724n_assn @ R2 ) )
      = ( ( ord_less_nat @ ( size_s6829681357464350627n_assn @ Ms ) @ ( size_s6829681357464350627n_assn @ Ns ) )
        | ( ( ( size_s6829681357464350627n_assn @ Ms )
            = ( size_s6829681357464350627n_assn @ Ns ) )
          & ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Ms @ Ns ) @ ( lenlex6671093235567761724n_assn @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4745_Cons__lenlex__iff,axiom,
    ! [M2: c,Ms: list_c,N: c,Ns: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ ( cons_c @ M2 @ Ms ) @ ( cons_c @ N @ Ns ) ) @ ( lenlex_c @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_c @ Ms ) @ ( size_size_list_c @ Ns ) )
        | ( ( ( size_size_list_c @ Ms )
            = ( size_size_list_c @ Ns ) )
          & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Ms @ Ns ) @ ( lenlex_c @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4746_Cons__lenlex__iff,axiom,
    ! [M2: d,Ms: list_d,N: d,Ns: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ ( cons_d @ M2 @ Ms ) @ ( cons_d @ N @ Ns ) ) @ ( lenlex_d @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_d @ Ms ) @ ( size_size_list_d @ Ns ) )
        | ( ( ( size_size_list_d @ Ms )
            = ( size_size_list_d @ Ns ) )
          & ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Ms @ Ns ) @ ( lenlex_d @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4747_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4748_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R2 ) )
        | ( ( M2 = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_4749_Cons__in__lex,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( lex_se2245640040323279819at_nat @ R2 ) )
      = ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
          & ( ( size_s8736152011456118867at_nat @ Xs )
            = ( size_s8736152011456118867at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs @ Ys ) @ ( lex_se2245640040323279819at_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4750_Cons__in__lex,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) @ ( lex_Pr7559194461235888923n_assn @ R2 ) )
      = ( ( ( member4743710847862083984n_assn @ ( produc6128706242671166167n_assn @ X @ Y ) @ R2 )
          & ( ( size_s6829681357464350627n_assn @ Xs )
            = ( size_s6829681357464350627n_assn @ Ys ) ) )
        | ( ( X = Y )
          & ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ Ys ) @ ( lex_Pr7559194461235888923n_assn @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4751_Cons__in__lex,axiom,
    ! [X: c,Xs: list_c,Y: c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ ( cons_c @ X @ Xs ) @ ( cons_c @ Y @ Ys ) ) @ ( lex_c @ R2 ) )
      = ( ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X @ Y ) @ R2 )
          & ( ( size_size_list_c @ Xs )
            = ( size_size_list_c @ Ys ) ) )
        | ( ( X = Y )
          & ( member7012247553653173456list_c @ ( produc5515445818230282391list_c @ Xs @ Ys ) @ ( lex_c @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4752_Cons__in__lex,axiom,
    ! [X: d,Xs: list_d,Y: d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ ( cons_d @ X @ Xs ) @ ( cons_d @ Y @ Ys ) ) @ ( lex_d @ R2 ) )
      = ( ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X @ Y ) @ R2 )
          & ( ( size_size_list_d @ Xs )
            = ( size_size_list_d @ Ys ) ) )
        | ( ( X = Y )
          & ( member1810801192463332112list_d @ ( produc242965421297323863list_d @ Xs @ Ys ) @ ( lex_d @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4753_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4754_Cons__in__lex,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R2 ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
          & ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_4755_list__assn_Oelims,axiom,
    ! [X: int > int > assn,Xa: list_int,Xb: list_int,Y: assn] :
      ( ( ( vEBT_L74593716426352029nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb
                    = ( cons_int @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L74593716426352029nt_int @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: int,Va: list_int] :
                  ( Xa
                  = ( cons_int @ V3 @ Va ) )
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: int,Va: list_int] :
                      ( Xb
                      = ( cons_int @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4756_list__assn_Oelims,axiom,
    ! [X: int > nat > assn,Xa: list_int,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L77084186935402305nt_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L77084186935402305nt_nat @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: int,Va: list_int] :
                  ( Xa
                  = ( cons_int @ V3 @ Va ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: nat,Va: list_nat] :
                      ( Xb
                      = ( cons_nat @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4757_list__assn_Oelims,axiom,
    ! [X: int > produc6575502325842934193n_assn > assn,Xa: list_int,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( ( vEBT_L1620274777744359680n_assn @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb
                    = ( cons_P2971678138204555879n_assn @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L1620274777744359680n_assn @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: int,Va: list_int] :
                  ( Xa
                  = ( cons_int @ V3 @ Va ) )
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                      ( Xb
                      = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4758_list__assn_Oelims,axiom,
    ! [X: nat > int > assn,Xa: list_nat,Xb: list_int,Y: assn] :
      ( ( ( vEBT_L8298612041380073281at_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb
                    = ( cons_int @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L8298612041380073281at_int @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] :
                  ( Xa
                  = ( cons_nat @ V3 @ Va ) )
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: int,Va: list_int] :
                      ( Xb
                      = ( cons_int @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4759_list__assn_Oelims,axiom,
    ! [X: nat > nat > assn,Xa: list_nat,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L8301102511889123557at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L8301102511889123557at_nat @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] :
                  ( Xa
                  = ( cons_nat @ V3 @ Va ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: nat,Va: list_nat] :
                      ( Xb
                      = ( cons_nat @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4760_list__assn_Oelims,axiom,
    ! [X: nat > produc6575502325842934193n_assn > assn,Xa: list_nat,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( ( vEBT_L9083610150846024284n_assn @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb
                    = ( cons_P2971678138204555879n_assn @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L9083610150846024284n_assn @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] :
                  ( Xa
                  = ( cons_nat @ V3 @ Va ) )
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                      ( Xb
                      = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4761_list__assn_Oelims,axiom,
    ! [X: produc6575502325842934193n_assn > int > assn,Xa: list_P8527749157015355191n_assn,Xb: list_int,Y: assn] :
      ( ( ( vEBT_L4760345170377931482sn_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa
                = ( cons_P2971678138204555879n_assn @ A4 @ As ) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb
                    = ( cons_int @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L4760345170377931482sn_int @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                  ( Xa
                  = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: int,Va: list_int] :
                      ( Xb
                      = ( cons_int @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4762_list__assn_Oelims,axiom,
    ! [X: produc6575502325842934193n_assn > nat > assn,Xa: list_P8527749157015355191n_assn,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L4762835640886981758sn_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa
                = ( cons_P2971678138204555879n_assn @ A4 @ As ) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L4762835640886981758sn_nat @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                  ( Xa
                  = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: nat,Va: list_nat] :
                      ( Xb
                      = ( cons_nat @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4763_list__assn_Oelims,axiom,
    ! [X: produc6575502325842934193n_assn > produc6575502325842934193n_assn > assn,Xa: list_P8527749157015355191n_assn,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( ( vEBT_L3108143461992071043n_assn @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa
                = ( cons_P2971678138204555879n_assn @ A4 @ As ) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb
                    = ( cons_P2971678138204555879n_assn @ C4 @ Cs2 ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C4 ) @ ( vEBT_L3108143461992071043n_assn @ X @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                  ( Xa
                  = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                      ( Xb
                      = ( cons_P2971678138204555879n_assn @ V3 @ Va ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_4764_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( count_list_int @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_4765_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( count_list_nat @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_4766_count__list_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( ( count_2530312006313534765n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_2530312006313534765n_assn @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_2530312006313534765n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Y )
          = ( count_2530312006313534765n_assn @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_4767_count__notin,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
     => ( ( count_1483299019173949923at_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4768_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( count_4203492906077236349at_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4769_count__notin,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ ( set_Pr400311997697478754_nat_o @ Xs ) )
     => ( ( count_5695430372156137289_nat_o @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4770_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ( count_list_int @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4771_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4772_count__notin,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs ) )
     => ( ( count_2530312006313534765n_assn @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_4773_power__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M2 ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_4774_power__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ M2 ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_4775_power__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_4776_power__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_4777_in__measures_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat > nat,Fs: list_s9130966667114977576at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ ( cons_s2538900923071588440at_nat @ F4 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_4778_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F4: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F4 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_4779_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_4780_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = nil_Pr2300489316682597567nt_int )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4781_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys )
        = nil_Pr6478340336191794275nt_nat )
      = ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4782_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_in7098856637274830732n_assn @ Xs @ Ys )
        = nil_Pr2440414457004260852n_assn )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4783_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( ( zip_nat_int @ Xs @ Ys )
        = nil_Pr1301135604781543011at_int )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4784_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4785_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_na5338819973521719528n_assn @ Xs @ Ys )
        = nil_Pr680377793251149648n_assn )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4786_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( ( zip_Pr1015554993053626726sn_int @ Xs @ Ys )
        = nil_Pr7991254345159696774sn_int )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4787_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys )
        = nil_Pr2945733327814117674sn_nat )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4788_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_Pr8806919808394484215n_assn @ Xs @ Ys )
        = nil_Pr4327835960982145133n_assn )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4789_zip__eq__Nil__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( ( zip_assn_assn @ Xs @ Ys )
        = nil_Pr5671120429643327159n_assn )
      = ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_4790_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( zip_int_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4791_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( nil_Pr6478340336191794275nt_nat
        = ( zip_int_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4792_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr2440414457004260852n_assn
        = ( zip_in7098856637274830732n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4793_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( nil_Pr1301135604781543011at_int
        = ( zip_nat_int @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4794_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4795_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr680377793251149648n_assn
        = ( zip_na5338819973521719528n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4796_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( nil_Pr7991254345159696774sn_int
        = ( zip_Pr1015554993053626726sn_int @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4797_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( nil_Pr2945733327814117674sn_nat
        = ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4798_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr4327835960982145133n_assn
        = ( zip_Pr8806919808394484215n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4799_Nil__eq__zip__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( zip_assn_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_4800_zip__Nil,axiom,
    ! [Ys: list_assn] :
      ( ( zip_assn_assn @ nil_assn @ Ys )
      = nil_Pr5671120429643327159n_assn ) ).

% zip_Nil
thf(fact_4801_Nil__is__rev__conv,axiom,
    ! [Xs: list_int] :
      ( ( nil_int
        = ( rev_int @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_rev_conv
thf(fact_4802_Nil__is__rev__conv,axiom,
    ! [Xs: list_nat] :
      ( ( nil_nat
        = ( rev_nat @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_rev_conv
thf(fact_4803_Nil__is__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( rev_Pr4855572775806611735n_assn @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_rev_conv
thf(fact_4804_rev__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rev_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rev_is_Nil_conv
thf(fact_4805_rev__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rev_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rev_is_Nil_conv
thf(fact_4806_rev__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( rev_Pr4855572775806611735n_assn @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rev_is_Nil_conv
thf(fact_4807_list__update__nonempty,axiom,
    ! [Xs: list_int,K3: nat,X: int] :
      ( ( ( list_update_int @ Xs @ K3 @ X )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% list_update_nonempty
thf(fact_4808_list__update__nonempty,axiom,
    ! [Xs: list_nat,K3: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K3 @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_4809_list__update__nonempty,axiom,
    ! [Xs: list_P8527749157015355191n_assn,K3: nat,X: produc6575502325842934193n_assn] :
      ( ( ( list_u4534839942911652127n_assn @ Xs @ K3 @ X )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% list_update_nonempty
thf(fact_4810_list__update__nonempty,axiom,
    ! [Xs: list_c,K3: nat,X: c] :
      ( ( ( list_update_c @ Xs @ K3 @ X )
        = nil_c )
      = ( Xs = nil_c ) ) ).

% list_update_nonempty
thf(fact_4811_list__update__nonempty,axiom,
    ! [Xs: list_d,K3: nat,X: d] :
      ( ( ( list_update_d @ Xs @ K3 @ X )
        = nil_d )
      = ( Xs = nil_d ) ) ).

% list_update_nonempty
thf(fact_4812_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_int @ N @ nil_int )
      = nil_Pr1301135604781543011at_int ) ).

% enumerate_simps(1)
thf(fact_4813_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_4814_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumer9200277086812776726n_assn @ N @ nil_Pr5671120429643327159n_assn )
      = nil_Pr680377793251149648n_assn ) ).

% enumerate_simps(1)
thf(fact_4815_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rotate1_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
thf(fact_4816_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_4817_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( rotate328796349445179396n_assn @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
thf(fact_4818_list__ex__simps_I2_J,axiom,
    ! [P: int > $o] :
      ~ ( list_ex_int @ P @ nil_int ) ).

% list_ex_simps(2)
thf(fact_4819_list__ex__simps_I2_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex_nat @ P @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_4820_list__ex__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > $o] :
      ~ ( list_e1087220904399170805n_assn @ P @ nil_Pr5671120429643327159n_assn ) ).

% list_ex_simps(2)
thf(fact_4821_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_int @ Begin @ End @ nil_int )
      = nil_int ) ).

% slice_Nil
thf(fact_4822_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_nat @ Begin @ End @ nil_nat )
      = nil_nat ) ).

% slice_Nil
thf(fact_4823_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_7964966981652229398n_assn @ Begin @ End @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_Nil
thf(fact_4824_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_int] :
      ( ( slice_int @ I @ I @ Xs )
      = nil_int ) ).

% slice_eq_bounds_empty
thf(fact_4825_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( slice_nat @ I @ I @ Xs )
      = nil_nat ) ).

% slice_eq_bounds_empty
thf(fact_4826_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( slice_7964966981652229398n_assn @ I @ I @ Xs )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_eq_bounds_empty
thf(fact_4827_in__measures_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ nil_se7398569966080822440at_nat ) ) ).

% in_measures(1)
thf(fact_4828_in__measures_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_4829_power__inject__exp,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ( power_power_rat @ A2 @ M2 )
          = ( power_power_rat @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_4830_power__inject__exp,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ( power_8256067586552552935nteger @ A2 @ M2 )
          = ( power_8256067586552552935nteger @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_4831_power__inject__exp,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ( power_power_nat @ A2 @ M2 )
          = ( power_power_nat @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_4832_power__inject__exp,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( power_power_int @ A2 @ M2 )
          = ( power_power_int @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_4833_set__empty,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( set_Pr1139785259514867910n_assn @ Xs )
        = bot_bo1176836662018730877n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% set_empty
thf(fact_4834_set__empty,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( ( set_Pr3765526544606949372at_nat @ Xs )
        = bot_bo228742789529271731at_nat )
      = ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty
thf(fact_4835_set__empty,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( ( set_Pr5648618587558075414at_nat @ Xs )
        = bot_bo2099793752762293965at_nat )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty
thf(fact_4836_set__empty,axiom,
    ! [Xs: list_o] :
      ( ( ( set_o2 @ Xs )
        = bot_bot_set_o )
      = ( Xs = nil_o ) ) ).

% set_empty
thf(fact_4837_set__empty,axiom,
    ! [Xs: list_nat] :
      ( ( ( set_nat2 @ Xs )
        = bot_bot_set_nat )
      = ( Xs = nil_nat ) ) ).

% set_empty
thf(fact_4838_set__empty,axiom,
    ! [Xs: list_int] :
      ( ( ( set_int2 @ Xs )
        = bot_bot_set_int )
      = ( Xs = nil_int ) ) ).

% set_empty
thf(fact_4839_set__empty2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( bot_bo1176836662018730877n_assn
        = ( set_Pr1139785259514867910n_assn @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% set_empty2
thf(fact_4840_set__empty2,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( set_Pr3765526544606949372at_nat @ Xs ) )
      = ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty2
thf(fact_4841_set__empty2,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty2
thf(fact_4842_set__empty2,axiom,
    ! [Xs: list_o] :
      ( ( bot_bot_set_o
        = ( set_o2 @ Xs ) )
      = ( Xs = nil_o ) ) ).

% set_empty2
thf(fact_4843_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat
        = ( set_nat2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% set_empty2
thf(fact_4844_set__empty2,axiom,
    ! [Xs: list_int] :
      ( ( bot_bot_set_int
        = ( set_int2 @ Xs ) )
      = ( Xs = nil_int ) ) ).

% set_empty2
thf(fact_4845_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
thf(fact_4846_length__0__conv,axiom,
    ! [Xs: list_c] :
      ( ( ( size_size_list_c @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_c ) ) ).

% length_0_conv
thf(fact_4847_length__0__conv,axiom,
    ! [Xs: list_d] :
      ( ( ( size_size_list_d @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_d ) ) ).

% length_0_conv
thf(fact_4848_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_4849_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_int ) ) ).

% length_0_conv
thf(fact_4850_singleton__rev__conv,axiom,
    ! [X: int,Xs: list_int] :
      ( ( ( cons_int @ X @ nil_int )
        = ( rev_int @ Xs ) )
      = ( ( cons_int @ X @ nil_int )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_4851_singleton__rev__conv,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( ( cons_nat @ X @ nil_nat )
        = ( rev_nat @ Xs ) )
      = ( ( cons_nat @ X @ nil_nat )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_4852_singleton__rev__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn )
        = ( rev_Pr4855572775806611735n_assn @ Xs ) )
      = ( ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_4853_rev__singleton__conv,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( rev_int @ Xs )
        = ( cons_int @ X @ nil_int ) )
      = ( Xs
        = ( cons_int @ X @ nil_int ) ) ) ).

% rev_singleton_conv
thf(fact_4854_rev__singleton__conv,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( rev_nat @ Xs )
        = ( cons_nat @ X @ nil_nat ) )
      = ( Xs
        = ( cons_nat @ X @ nil_nat ) ) ) ).

% rev_singleton_conv
thf(fact_4855_rev__singleton__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ( rev_Pr4855572775806611735n_assn @ Xs )
        = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = ( Xs
        = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ) ).

% rev_singleton_conv
thf(fact_4856_Nil__lenlex__iff1,axiom,
    ! [Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ns ) @ ( lenlex_int @ R2 ) )
      = ( Ns != nil_int ) ) ).

% Nil_lenlex_iff1
thf(fact_4857_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R2 ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_4858_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ Ns ) @ ( lenlex6671093235567761724n_assn @ R2 ) )
      = ( Ns != nil_Pr5671120429643327159n_assn ) ) ).

% Nil_lenlex_iff1
thf(fact_4859_insert__Nil,axiom,
    ! [X: int] :
      ( ( insert_int @ X @ nil_int )
      = ( cons_int @ X @ nil_int ) ) ).

% insert_Nil
thf(fact_4860_insert__Nil,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ nil_nat )
      = ( cons_nat @ X @ nil_nat ) ) ).

% insert_Nil
thf(fact_4861_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( insert3246601298802261197n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ).

% insert_Nil
thf(fact_4862_list__assn__aux__simps_I1_J,axiom,
    ! [P: int > int > assn,L3: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ nil_int @ L3 )
      = ( pure_assn @ ( L3 = nil_int ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4863_list__assn__aux__simps_I1_J,axiom,
    ! [P: int > nat > assn,L3: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ nil_int @ L3 )
      = ( pure_assn @ ( L3 = nil_nat ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4864_list__assn__aux__simps_I1_J,axiom,
    ! [P: int > produc6575502325842934193n_assn > assn,L3: list_P8527749157015355191n_assn] :
      ( ( vEBT_L1620274777744359680n_assn @ P @ nil_int @ L3 )
      = ( pure_assn @ ( L3 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4865_list__assn__aux__simps_I1_J,axiom,
    ! [P: nat > int > assn,L3: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ nil_nat @ L3 )
      = ( pure_assn @ ( L3 = nil_int ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4866_list__assn__aux__simps_I1_J,axiom,
    ! [P: nat > nat > assn,L3: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ nil_nat @ L3 )
      = ( pure_assn @ ( L3 = nil_nat ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4867_list__assn__aux__simps_I1_J,axiom,
    ! [P: nat > produc6575502325842934193n_assn > assn,L3: list_P8527749157015355191n_assn] :
      ( ( vEBT_L9083610150846024284n_assn @ P @ nil_nat @ L3 )
      = ( pure_assn @ ( L3 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4868_list__assn__aux__simps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > int > assn,L3: list_int] :
      ( ( vEBT_L4760345170377931482sn_int @ P @ nil_Pr5671120429643327159n_assn @ L3 )
      = ( pure_assn @ ( L3 = nil_int ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4869_list__assn__aux__simps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > nat > assn,L3: list_nat] :
      ( ( vEBT_L4762835640886981758sn_nat @ P @ nil_Pr5671120429643327159n_assn @ L3 )
      = ( pure_assn @ ( L3 = nil_nat ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4870_list__assn__aux__simps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > assn,L3: list_P8527749157015355191n_assn] :
      ( ( vEBT_L3108143461992071043n_assn @ P @ nil_Pr5671120429643327159n_assn @ L3 )
      = ( pure_assn @ ( L3 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_4871_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_4872_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_4873_list__assn__aux__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > int > assn,L: list_P8527749157015355191n_assn] :
      ( ( vEBT_L4760345170377931482sn_int @ P @ L @ nil_int )
      = ( pure_assn @ ( L = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_4874_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_4875_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_4876_list__assn__aux__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > nat > assn,L: list_P8527749157015355191n_assn] :
      ( ( vEBT_L4762835640886981758sn_nat @ P @ L @ nil_nat )
      = ( pure_assn @ ( L = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_4877_list__assn__aux__simps_I2_J,axiom,
    ! [P: int > produc6575502325842934193n_assn > assn,L: list_int] :
      ( ( vEBT_L1620274777744359680n_assn @ P @ L @ nil_Pr5671120429643327159n_assn )
      = ( pure_assn @ ( L = nil_int ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_4878_list__assn__aux__simps_I2_J,axiom,
    ! [P: nat > produc6575502325842934193n_assn > assn,L: list_nat] :
      ( ( vEBT_L9083610150846024284n_assn @ P @ L @ nil_Pr5671120429643327159n_assn )
      = ( pure_assn @ ( L = nil_nat ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_4879_list__assn__aux__simps_I2_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > assn,L: list_P8527749157015355191n_assn] :
      ( ( vEBT_L3108143461992071043n_assn @ P @ L @ nil_Pr5671120429643327159n_assn )
      = ( pure_assn @ ( L = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_4880_power__strict__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_4881_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_4882_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_4883_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_4884_power__eq__0__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ( power_8256067586552552935nteger @ A2 @ N )
        = zero_z3403309356797280102nteger )
      = ( ( A2 = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4885_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ( power_power_rat @ A2 @ N )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4886_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( ( power_power_nat @ A2 @ N )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4887_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( ( power_power_int @ A2 @ N )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_4888_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s6829681357464350627n_assn @ Xs ) )
      = ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
thf(fact_4889_length__greater__0__conv,axiom,
    ! [Xs: list_c] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_c @ Xs ) )
      = ( Xs != nil_c ) ) ).

% length_greater_0_conv
thf(fact_4890_length__greater__0__conv,axiom,
    ! [Xs: list_d] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_d @ Xs ) )
      = ( Xs != nil_d ) ) ).

% length_greater_0_conv
thf(fact_4891_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_4892_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
      = ( Xs != nil_int ) ) ).

% length_greater_0_conv
thf(fact_4893_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M2 ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4894_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ M2 ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4895_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4896_power__strict__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_4897_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
            = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_4898_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) )
            = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_4899_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
            = ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_4900_power__mono__iff,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
            = ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_4901_power__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_4902_power__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_4903_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_4904_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_4905_length__ge__1__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s6829681357464350627n_assn @ L ) )
      = ( L != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
thf(fact_4906_length__ge__1__conv,axiom,
    ! [L: list_c] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_c @ L ) )
      = ( L != nil_c ) ) ).

% length_ge_1_conv
thf(fact_4907_length__ge__1__conv,axiom,
    ! [L: list_d] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_d @ L ) )
      = ( L != nil_d ) ) ).

% length_ge_1_conv
thf(fact_4908_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_4909_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_4910_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] :
            ( X
           != ( cons_list_int @ nil_int @ Xss ) )
       => ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
              ( X
             != ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_4911_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_4912_transpose_Ocases,axiom,
    ! [X: list_l6351802567095793725n_assn] :
      ( ( X != nil_li5476096274760905021n_assn )
     => ( ! [Xss: list_l6351802567095793725n_assn] :
            ( X
           != ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ Xss ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Xss: list_l6351802567095793725n_assn] :
              ( X
             != ( cons_l2423627976422276333n_assn @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_4913_Nil__lenlex__iff2,axiom,
    ! [Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ns @ nil_int ) @ ( lenlex_int @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_4914_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_4915_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Ns @ nil_Pr5671120429643327159n_assn ) @ ( lenlex6671093235567761724n_assn @ R2 ) ) ).

% Nil_lenlex_iff2
thf(fact_4916_Nil2__notin__lex,axiom,
    ! [Xs: list_int,R2: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ nil_int ) @ ( lex_int @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_4917_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_4918_Nil2__notin__lex,axiom,
    ! [Xs: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ Xs @ nil_Pr5671120429643327159n_assn ) @ ( lex_Pr7559194461235888923n_assn @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_4919_Nil__notin__lex,axiom,
    ! [Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ys ) @ ( lex_int @ R2 ) ) ).

% Nil_notin_lex
thf(fact_4920_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R2 ) ) ).

% Nil_notin_lex
thf(fact_4921_Nil__notin__lex,axiom,
    ! [Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ~ ( member3994142547374651152n_assn @ ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ Ys ) @ ( lex_Pr7559194461235888923n_assn @ R2 ) ) ).

% Nil_notin_lex
thf(fact_4922_zip_Osimps_I1_J,axiom,
    ! [Xs: list_assn] :
      ( ( zip_assn_assn @ Xs @ nil_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% zip.simps(1)
thf(fact_4923_mset__distrib,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,M5: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ M5 @ N7 ) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A
              = ( plus_p7104986032573967614at_nat @ Am @ An ) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B
                  = ( plus_p7104986032573967614at_nat @ Bm @ Bn ) )
               => ( ( M5
                    = ( plus_p7104986032573967614at_nat @ Am @ Bm ) )
                 => ( N7
                   != ( plus_p7104986032573967614at_nat @ An @ Bn ) ) ) ) ) ) ).

% mset_distrib
thf(fact_4924_count__list_Osimps_I1_J,axiom,
    ! [Y: int] :
      ( ( count_list_int @ nil_int @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_4925_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] :
      ( ( count_list_nat @ nil_nat @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_4926_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] :
      ( ( count_2530312006313534765n_assn @ nil_Pr5671120429643327159n_assn @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_4927_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] :
      ( ( product_assn_assn @ nil_assn @ Uu )
      = nil_Pr5671120429643327159n_assn ) ).

% product.simps(1)
thf(fact_4928_power__commutes,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_4929_power__commutes,axiom,
    ! [A2: assn,N: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ A2 )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_4930_power__commutes,axiom,
    ! [A2: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_4931_power__commutes,axiom,
    ! [A2: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_4932_power__commutes,axiom,
    ! [A2: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_4933_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_4934_power__mult__distrib,axiom,
    ! [A2: assn,B2: assn,N: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A2 @ B2 ) @ N )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ ( power_power_assn @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_4935_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A2 @ B2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_4936_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_4937_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A2 @ B2 ) @ N )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_4938_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4939_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4940_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4941_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4942_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_4943_successively_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P5: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P5 @ nil_int ) )
     => ( ! [P5: int > int > $o,X3: int] :
            ( X
           != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X3 @ nil_int ) ) )
       => ~ ! [P5: int > int > $o,X3: int,Y2: int,Xs2: list_int] :
              ( X
             != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_4944_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X3: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X3: nat,Y2: nat,Xs2: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_4945_successively_Ocases,axiom,
    ! [X: produc4530688714081574576n_assn] :
      ( ! [P5: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc6115016301897879072n_assn @ P5 @ nil_Pr5671120429643327159n_assn ) )
     => ( ! [P5: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn] :
            ( X
           != ( produc6115016301897879072n_assn @ P5 @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) ) )
       => ~ ! [P5: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( X
             != ( produc6115016301897879072n_assn @ P5 @ ( cons_P2971678138204555879n_assn @ X3 @ ( cons_P2971678138204555879n_assn @ Y2 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_4946_sorted__wrt_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P5: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P5 @ nil_int ) )
     => ~ ! [P5: int > int > $o,X3: int,Ys5: list_int] :
            ( X
           != ( produc8618682346314911123st_int @ P5 @ ( cons_int @ X3 @ Ys5 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_4947_sorted__wrt_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ~ ! [P5: nat > nat > $o,X3: nat,Ys5: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ Ys5 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_4948_sorted__wrt_Ocases,axiom,
    ! [X: produc4530688714081574576n_assn] :
      ( ! [P5: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc6115016301897879072n_assn @ P5 @ nil_Pr5671120429643327159n_assn ) )
     => ~ ! [P5: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( X
           != ( produc6115016301897879072n_assn @ P5 @ ( cons_P2971678138204555879n_assn @ X3 @ Ys5 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_4949_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X3: int,Xs2: list_int] :
            ( L
           != ( cons_int @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_4950_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( L
           != ( cons_nat @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_4951_neq__NilE,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( L
           != ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_4952_list__2pre__induct,axiom,
    ! [P: list_int > list_int > $o,W1: list_int,W2: list_int] :
      ( ( P @ nil_int @ nil_int )
     => ( ! [E2: int,W12: list_int,W22: list_int] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_int @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: int,W13: list_int,W23: list_int] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_int @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4953_list__2pre__induct,axiom,
    ! [P: list_int > list_nat > $o,W1: list_int,W2: list_nat] :
      ( ( P @ nil_int @ nil_nat )
     => ( ! [E2: int,W12: list_int,W22: list_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_int @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: nat,W13: list_int,W23: list_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_nat @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4954_list__2pre__induct,axiom,
    ! [P: list_int > list_P8527749157015355191n_assn > $o,W1: list_int,W2: list_P8527749157015355191n_assn] :
      ( ( P @ nil_int @ nil_Pr5671120429643327159n_assn )
     => ( ! [E2: int,W12: list_int,W22: list_P8527749157015355191n_assn] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_int @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_int,W23: list_P8527749157015355191n_assn] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_P2971678138204555879n_assn @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4955_list__2pre__induct,axiom,
    ! [P: list_nat > list_int > $o,W1: list_nat,W2: list_int] :
      ( ( P @ nil_nat @ nil_int )
     => ( ! [E2: nat,W12: list_nat,W22: list_int] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_nat @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: int,W13: list_nat,W23: list_int] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_int @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4956_list__2pre__induct,axiom,
    ! [P: list_nat > list_nat > $o,W1: list_nat,W2: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [E2: nat,W12: list_nat,W22: list_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_nat @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: nat,W13: list_nat,W23: list_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_nat @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4957_list__2pre__induct,axiom,
    ! [P: list_nat > list_P8527749157015355191n_assn > $o,W1: list_nat,W2: list_P8527749157015355191n_assn] :
      ( ( P @ nil_nat @ nil_Pr5671120429643327159n_assn )
     => ( ! [E2: nat,W12: list_nat,W22: list_P8527749157015355191n_assn] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_nat @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_nat,W23: list_P8527749157015355191n_assn] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_P2971678138204555879n_assn @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4958_list__2pre__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_int > $o,W1: list_P8527749157015355191n_assn,W2: list_int] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_int )
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_int] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_P2971678138204555879n_assn @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: int,W13: list_P8527749157015355191n_assn,W23: list_int] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_int @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4959_list__2pre__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_nat > $o,W1: list_P8527749157015355191n_assn,W2: list_nat] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_nat )
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_P2971678138204555879n_assn @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: nat,W13: list_P8527749157015355191n_assn,W23: list_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_nat @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4960_list__2pre__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,W1: list_P8527749157015355191n_assn,W2: list_P8527749157015355191n_assn] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_Pr5671120429643327159n_assn )
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_P8527749157015355191n_assn] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_P2971678138204555879n_assn @ E2 @ W12 ) @ W22 ) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_P8527749157015355191n_assn,W23: list_P8527749157015355191n_assn] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_P2971678138204555879n_assn @ E2 @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_4961_list__induct__first2,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ( P @ nil_int )
     => ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X12: int,X23: int,Xs2: list_int] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_int @ X12 @ ( cons_int @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_4962_list__induct__first2,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X12: nat,X23: nat,Xs2: list_nat] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_nat @ X12 @ ( cons_nat @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_4963_list__induct__first2,axiom,
    ! [P: list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn] :
      ( ( P @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X12: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_P2971678138204555879n_assn @ X12 @ ( cons_P2971678138204555879n_assn @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_4964_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_int > list_int > $o,R: int > int > $o,Xs: list_int,Ys: list_int] :
      ( ! [Xs2: list_int] : ( P @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_int @ Y2 @ Ys5 ) )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_int @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4965_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_nat > list_int > $o,R: nat > int > $o,Xs: list_nat,Ys: list_int] :
      ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_int @ Y2 @ Ys5 ) )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_nat @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4966_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_int > $o,R: produc6575502325842934193n_assn > int > $o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_int @ Y2 @ Ys5 ) )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4967_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_int > list_nat > $o,R: int > nat > $o,Xs: list_int,Ys: list_nat] :
      ( ! [Xs2: list_int] : ( P @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_nat @ Y2 @ Ys5 ) )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_int @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4968_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_nat > list_nat > $o,R: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_nat @ Y2 @ Ys5 ) )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_nat @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4969_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_nat > $o,R: produc6575502325842934193n_assn > nat > $o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_nat @ Y2 @ Ys5 ) )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4970_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_int > list_P8527749157015355191n_assn > $o,R: int > produc6575502325842934193n_assn > $o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_int] : ( P @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_int @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4971_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_nat > list_P8527749157015355191n_assn > $o,R: nat > produc6575502325842934193n_assn > $o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_nat @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4972_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,R: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y2 )
               => ( ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys5 )
                 => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_4973_list_Odistinct_I1_J,axiom,
    ! [X21: int,X222: list_int] :
      ( nil_int
     != ( cons_int @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_4974_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_4975_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
      ( nil_Pr5671120429643327159n_assn
     != ( cons_P2971678138204555879n_assn @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_4976_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X222: list_int] :
      ( ( List
        = ( cons_int @ X21 @ X222 ) )
     => ( List != nil_int ) ) ).

% list.discI
thf(fact_4977_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_4978_list_OdiscI,axiom,
    ! [List: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
      ( ( List
        = ( cons_P2971678138204555879n_assn @ X21 @ X222 ) )
     => ( List != nil_Pr5671120429643327159n_assn ) ) ).

% list.discI
thf(fact_4979_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X223: list_int] :
            ( Y
           != ( cons_int @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_4980_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_4981_list_Oexhaust,axiom,
    ! [Y: list_P8527749157015355191n_assn] :
      ( ( Y != nil_Pr5671120429643327159n_assn )
     => ~ ! [X212: produc6575502325842934193n_assn,X223: list_P8527749157015355191n_assn] :
            ( Y
           != ( cons_P2971678138204555879n_assn @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_4982_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X3: int,Xs2: list_int] :
          ( X
         != ( cons_int @ X3 @ Xs2 ) )
     => ( X = nil_int ) ) ).

% min_list.cases
thf(fact_4983_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs2: list_nat] :
          ( X
         != ( cons_nat @ X3 @ Xs2 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_4984_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X3: int] :
            ( X
           != ( cons_int @ X3 @ nil_int ) )
       => ~ ! [X3: int,Y2: int,Xs2: list_int] :
              ( X
             != ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_4985_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] :
            ( X
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
              ( X
             != ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_4986_remdups__adj_Ocases,axiom,
    ! [X: list_P8527749157015355191n_assn] :
      ( ( X != nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( X
           != ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( X
             != ( cons_P2971678138204555879n_assn @ X3 @ ( cons_P2971678138204555879n_assn @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_4987_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
      = ( ? [Y5: int,Ys3: list_int] :
            ( Xs
            = ( cons_int @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_4988_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_4989_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
      = ( ? [Y5: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( Xs
            = ( cons_P2971678138204555879n_assn @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_4990_list__induct2_H,axiom,
    ! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
      ( ( P @ nil_int @ nil_int )
     => ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y2: int,Ys5: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys5 ) )
         => ( ! [X3: int,Xs2: list_int,Y2: int,Ys5: list_int] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4991_list__induct2_H,axiom,
    ! [P: list_int > list_nat > $o,Xs: list_int,Ys: list_nat] :
      ( ( P @ nil_int @ nil_nat )
     => ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys5: list_nat] : ( P @ nil_int @ ( cons_nat @ Y2 @ Ys5 ) )
         => ( ! [X3: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4992_list__induct2_H,axiom,
    ! [P: list_int > list_P8527749157015355191n_assn > $o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( P @ nil_int @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : ( P @ nil_int @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
         => ( ! [X3: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4993_list__induct2_H,axiom,
    ! [P: list_nat > list_int > $o,Xs: list_nat,Ys: list_int] :
      ( ( P @ nil_nat @ nil_int )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y2: int,Ys5: list_int] : ( P @ nil_nat @ ( cons_int @ Y2 @ Ys5 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4994_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys5: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys5 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4995_list__induct2_H,axiom,
    ! [P: list_nat > list_P8527749157015355191n_assn > $o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( P @ nil_nat @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : ( P @ nil_nat @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4996_list__induct2_H,axiom,
    ! [P: list_P8527749157015355191n_assn > list_int > $o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_int )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y2: int,Ys5: list_int] : ( P @ nil_Pr5671120429643327159n_assn @ ( cons_int @ Y2 @ Ys5 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4997_list__induct2_H,axiom,
    ! [P: list_P8527749157015355191n_assn > list_nat > $o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_nat )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys5: list_nat] : ( P @ nil_Pr5671120429643327159n_assn @ ( cons_nat @ Y2 @ Ys5 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4998_list__induct2_H,axiom,
    ! [P: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( P @ nil_Pr5671120429643327159n_assn @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : ( P @ nil_Pr5671120429643327159n_assn @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y2 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_4999_list__nonempty__induct,axiom,
    ! [Xs: list_int,P: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X3: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_int @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_5000_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_5001_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: list_P8527749157015355191n_assn > $o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_5002_distinct_Osimps_I1_J,axiom,
    distinct_int @ nil_int ).

% distinct.simps(1)
thf(fact_5003_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_5004_distinct_Osimps_I1_J,axiom,
    distin4809818325527169450n_assn @ nil_Pr5671120429643327159n_assn ).

% distinct.simps(1)
thf(fact_5005_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] : ( sorted3986126766855596574n_assn @ P @ nil_Pr5671120429643327159n_assn ) ).

% sorted_wrt.simps(1)
thf(fact_5006_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).

% sorted_wrt.simps(1)
thf(fact_5007_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: int > int > $o] : ( sorted_wrt_int @ P @ nil_int ) ).

% sorted_wrt.simps(1)
thf(fact_5008_rev_Osimps_I1_J,axiom,
    ( ( rev_int @ nil_int )
    = nil_int ) ).

% rev.simps(1)
thf(fact_5009_rev_Osimps_I1_J,axiom,
    ( ( rev_nat @ nil_nat )
    = nil_nat ) ).

% rev.simps(1)
thf(fact_5010_rev_Osimps_I1_J,axiom,
    ( ( rev_Pr4855572775806611735n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% rev.simps(1)
thf(fact_5011_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: int] :
      ( ( list_update_int @ nil_int @ I @ V2 )
      = nil_int ) ).

% list_update.simps(1)
thf(fact_5012_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V2 )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_5013_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: produc6575502325842934193n_assn] :
      ( ( list_u4534839942911652127n_assn @ nil_Pr5671120429643327159n_assn @ I @ V2 )
      = nil_Pr5671120429643327159n_assn ) ).

% list_update.simps(1)
thf(fact_5014_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: c] :
      ( ( list_update_c @ nil_c @ I @ V2 )
      = nil_c ) ).

% list_update.simps(1)
thf(fact_5015_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: d] :
      ( ( list_update_d @ nil_d @ I @ V2 )
      = nil_d ) ).

% list_update.simps(1)
thf(fact_5016_list__update__code_I1_J,axiom,
    ! [I: nat,Y: int] :
      ( ( list_update_int @ nil_int @ I @ Y )
      = nil_int ) ).

% list_update_code(1)
thf(fact_5017_list__update__code_I1_J,axiom,
    ! [I: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_5018_list__update__code_I1_J,axiom,
    ! [I: nat,Y: produc6575502325842934193n_assn] :
      ( ( list_u4534839942911652127n_assn @ nil_Pr5671120429643327159n_assn @ I @ Y )
      = nil_Pr5671120429643327159n_assn ) ).

% list_update_code(1)
thf(fact_5019_list__update__code_I1_J,axiom,
    ! [I: nat,Y: c] :
      ( ( list_update_c @ nil_c @ I @ Y )
      = nil_c ) ).

% list_update_code(1)
thf(fact_5020_list__update__code_I1_J,axiom,
    ! [I: nat,Y: d] :
      ( ( list_update_d @ nil_d @ I @ Y )
      = nil_d ) ).

% list_update_code(1)
thf(fact_5021_butlast_Osimps_I1_J,axiom,
    ( ( butlast_int @ nil_int )
    = nil_int ) ).

% butlast.simps(1)
thf(fact_5022_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_5023_butlast_Osimps_I1_J,axiom,
    ( ( butlas3012047794866324995n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% butlast.simps(1)
thf(fact_5024_remove1_Osimps_I1_J,axiom,
    ! [X: int] :
      ( ( remove1_int @ X @ nil_int )
      = nil_int ) ).

% remove1.simps(1)
thf(fact_5025_remove1_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( remove1_nat @ X @ nil_nat )
      = nil_nat ) ).

% remove1.simps(1)
thf(fact_5026_remove1_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( remove1670527618125605709n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% remove1.simps(1)
thf(fact_5027_removeAll_Osimps_I1_J,axiom,
    ! [X: int] :
      ( ( removeAll_int @ X @ nil_int )
      = nil_int ) ).

% removeAll.simps(1)
thf(fact_5028_removeAll_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( removeAll_nat @ X @ nil_nat )
      = nil_nat ) ).

% removeAll.simps(1)
thf(fact_5029_removeAll_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( remove1763081955827946941n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% removeAll.simps(1)
thf(fact_5030_merge_Osimps_I1_J,axiom,
    ! [L23: list_int] :
      ( ( merge_int @ nil_int @ L23 )
      = L23 ) ).

% merge.simps(1)
thf(fact_5031_merge_Osimps_I1_J,axiom,
    ! [L23: list_nat] :
      ( ( merge_nat @ nil_nat @ L23 )
      = L23 ) ).

% merge.simps(1)
thf(fact_5032_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_int @ nil_int )
    = nil_int ) ).

% rotate1.simps(1)
thf(fact_5033_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_5034_rotate1_Osimps_I1_J,axiom,
    ( ( rotate328796349445179396n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% rotate1.simps(1)
thf(fact_5035_zipf_Osimps_I1_J,axiom,
    ! [F4: int > int > int] :
      ( ( zipf_int_int_int @ F4 @ nil_int @ nil_int )
      = nil_int ) ).

% zipf.simps(1)
thf(fact_5036_zipf_Osimps_I1_J,axiom,
    ! [F4: int > int > nat] :
      ( ( zipf_int_int_nat @ F4 @ nil_int @ nil_int )
      = nil_nat ) ).

% zipf.simps(1)
thf(fact_5037_zipf_Osimps_I1_J,axiom,
    ! [F4: int > nat > int] :
      ( ( zipf_int_nat_int @ F4 @ nil_int @ nil_nat )
      = nil_int ) ).

% zipf.simps(1)
thf(fact_5038_zipf_Osimps_I1_J,axiom,
    ! [F4: int > nat > nat] :
      ( ( zipf_int_nat_nat @ F4 @ nil_int @ nil_nat )
      = nil_nat ) ).

% zipf.simps(1)
thf(fact_5039_zipf_Osimps_I1_J,axiom,
    ! [F4: nat > int > int] :
      ( ( zipf_nat_int_int @ F4 @ nil_nat @ nil_int )
      = nil_int ) ).

% zipf.simps(1)
thf(fact_5040_zipf_Osimps_I1_J,axiom,
    ! [F4: nat > int > nat] :
      ( ( zipf_nat_int_nat @ F4 @ nil_nat @ nil_int )
      = nil_nat ) ).

% zipf.simps(1)
thf(fact_5041_zipf_Osimps_I1_J,axiom,
    ! [F4: nat > nat > int] :
      ( ( zipf_nat_nat_int @ F4 @ nil_nat @ nil_nat )
      = nil_int ) ).

% zipf.simps(1)
thf(fact_5042_zipf_Osimps_I1_J,axiom,
    ! [F4: nat > nat > nat] :
      ( ( zipf_nat_nat_nat @ F4 @ nil_nat @ nil_nat )
      = nil_nat ) ).

% zipf.simps(1)
thf(fact_5043_zipf_Osimps_I1_J,axiom,
    ! [F4: int > int > produc6575502325842934193n_assn] :
      ( ( zipf_i7756021159461276697n_assn @ F4 @ nil_int @ nil_int )
      = nil_Pr5671120429643327159n_assn ) ).

% zipf.simps(1)
thf(fact_5044_zipf_Osimps_I1_J,axiom,
    ! [F4: int > nat > produc6575502325842934193n_assn] :
      ( ( zipf_i5995984495708165493n_assn @ F4 @ nil_int @ nil_nat )
      = nil_Pr5671120429643327159n_assn ) ).

% zipf.simps(1)
thf(fact_5045_null__rec_I2_J,axiom,
    null_int @ nil_int ).

% null_rec(2)
thf(fact_5046_null__rec_I2_J,axiom,
    null_nat @ nil_nat ).

% null_rec(2)
thf(fact_5047_null__rec_I2_J,axiom,
    null_P7596217980778516671n_assn @ nil_Pr5671120429643327159n_assn ).

% null_rec(2)
thf(fact_5048_eq__Nil__null,axiom,
    ! [Xs: list_int] :
      ( ( Xs = nil_int )
      = ( null_int @ Xs ) ) ).

% eq_Nil_null
thf(fact_5049_eq__Nil__null,axiom,
    ! [Xs: list_nat] :
      ( ( Xs = nil_nat )
      = ( null_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_5050_eq__Nil__null,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
      = ( null_P7596217980778516671n_assn @ Xs ) ) ).

% eq_Nil_null
thf(fact_5051_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_5052_zero__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_5053_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_5054_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_5055_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_5056_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_5057_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_5058_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_5059_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_5060_power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_5061_power__Suc,axiom,
    ! [A2: assn,N: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N ) )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_5062_power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_5063_power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_5064_power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_5065_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_5066_power__Suc2,axiom,
    ! [A2: assn,N: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_5067_power__Suc2,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_5068_power__Suc2,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_5069_power__Suc2,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_5070_power__add,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_add
thf(fact_5071_power__add,axiom,
    ! [A2: assn,M2: nat,N: nat] :
      ( ( power_power_assn @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ M2 ) @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_add
thf(fact_5072_power__add,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ M2 ) @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_add
thf(fact_5073_power__add,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_add
thf(fact_5074_power__add,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( power_power_int @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) ) ) ).

% power_add
thf(fact_5075_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_5076_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_c,P: list_c > list_c > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( P @ nil_c @ nil_c )
       => ( ! [X3: c,Xs2: list_c,Y2: c,Ys5: list_c] :
              ( ( ( size_size_list_c @ Xs2 )
                = ( size_size_list_c @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5077_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_d,P: list_c > list_d > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( P @ nil_c @ nil_d )
       => ( ! [X3: c,Xs2: list_c,Y2: d,Ys5: list_d] :
              ( ( ( size_size_list_c @ Xs2 )
                = ( size_size_list_d @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5078_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_nat,P: list_c > list_nat > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_c @ nil_nat )
       => ( ! [X3: c,Xs2: list_c,Y2: nat,Ys5: list_nat] :
              ( ( ( size_size_list_c @ Xs2 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5079_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_int,P: list_c > list_int > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_c @ nil_int )
       => ( ! [X3: c,Xs2: list_c,Y2: int,Ys5: list_int] :
              ( ( ( size_size_list_c @ Xs2 )
                = ( size_size_list_int @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5080_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_c,P: list_d > list_c > $o] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( P @ nil_d @ nil_c )
       => ( ! [X3: d,Xs2: list_d,Y2: c,Ys5: list_c] :
              ( ( ( size_size_list_d @ Xs2 )
                = ( size_size_list_c @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_d @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5081_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_d,P: list_d > list_d > $o] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( P @ nil_d @ nil_d )
       => ( ! [X3: d,Xs2: list_d,Y2: d,Ys5: list_d] :
              ( ( ( size_size_list_d @ Xs2 )
                = ( size_size_list_d @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_d @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5082_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_nat,P: list_d > list_nat > $o] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_d @ nil_nat )
       => ( ! [X3: d,Xs2: list_d,Y2: nat,Ys5: list_nat] :
              ( ( ( size_size_list_d @ Xs2 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_d @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5083_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_int,P: list_d > list_int > $o] :
      ( ( ( size_size_list_d @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P @ nil_d @ nil_int )
       => ( ! [X3: d,Xs2: list_d,Y2: int,Ys5: list_int] :
              ( ( ( size_size_list_d @ Xs2 )
                = ( size_size_list_int @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_d @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5084_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_c,P: list_nat > list_c > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( P @ nil_nat @ nil_c )
       => ( ! [X3: nat,Xs2: list_nat,Y2: c,Ys5: list_c] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_c @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5085_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_d,P: list_nat > list_d > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( P @ nil_nat @ nil_d )
       => ( ! [X3: nat,Xs2: list_nat,Y2: d,Ys5: list_d] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_d @ Ys5 ) )
             => ( ( P @ Xs2 @ Ys5 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_5086_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_c,P: list_c > list_c > list_c > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( ( size_size_list_c @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( P @ nil_c @ nil_c @ nil_c )
         => ( ! [X3: c,Xs2: list_c,Y2: c,Ys5: list_c,Z4: c,Zs2: list_c] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_c @ Ys5 ) )
               => ( ( ( size_size_list_c @ Ys5 )
                    = ( size_size_list_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5087_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_d,P: list_c > list_c > list_d > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( ( size_size_list_c @ Ys )
          = ( size_size_list_d @ Zs ) )
       => ( ( P @ nil_c @ nil_c @ nil_d )
         => ( ! [X3: c,Xs2: list_c,Y2: c,Ys5: list_c,Z4: d,Zs2: list_d] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_c @ Ys5 ) )
               => ( ( ( size_size_list_c @ Ys5 )
                    = ( size_size_list_d @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) @ ( cons_d @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5088_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_nat,P: list_c > list_c > list_nat > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( ( size_size_list_c @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_c @ nil_c @ nil_nat )
         => ( ! [X3: c,Xs2: list_c,Y2: c,Ys5: list_c,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_c @ Ys5 ) )
               => ( ( ( size_size_list_c @ Ys5 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5089_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_int,P: list_c > list_c > list_int > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_c @ Ys ) )
     => ( ( ( size_size_list_c @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_c @ nil_c @ nil_int )
         => ( ! [X3: c,Xs2: list_c,Y2: c,Ys5: list_c,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_c @ Ys5 ) )
               => ( ( ( size_size_list_c @ Ys5 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_c @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5090_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_c,P: list_c > list_d > list_c > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( ( size_size_list_d @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( P @ nil_c @ nil_d @ nil_c )
         => ( ! [X3: c,Xs2: list_c,Y2: d,Ys5: list_d,Z4: c,Zs2: list_c] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_d @ Ys5 ) )
               => ( ( ( size_size_list_d @ Ys5 )
                    = ( size_size_list_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5091_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_d,P: list_c > list_d > list_d > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( ( size_size_list_d @ Ys )
          = ( size_size_list_d @ Zs ) )
       => ( ( P @ nil_c @ nil_d @ nil_d )
         => ( ! [X3: c,Xs2: list_c,Y2: d,Ys5: list_d,Z4: d,Zs2: list_d] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_d @ Ys5 ) )
               => ( ( ( size_size_list_d @ Ys5 )
                    = ( size_size_list_d @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) @ ( cons_d @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5092_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_nat,P: list_c > list_d > list_nat > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( ( size_size_list_d @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_c @ nil_d @ nil_nat )
         => ( ! [X3: c,Xs2: list_c,Y2: d,Ys5: list_d,Z4: nat,Zs2: list_nat] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_d @ Ys5 ) )
               => ( ( ( size_size_list_d @ Ys5 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) @ ( cons_nat @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5093_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_int,P: list_c > list_d > list_int > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_d @ Ys ) )
     => ( ( ( size_size_list_d @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P @ nil_c @ nil_d @ nil_int )
         => ( ! [X3: c,Xs2: list_c,Y2: d,Ys5: list_d,Z4: int,Zs2: list_int] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_d @ Ys5 ) )
               => ( ( ( size_size_list_d @ Ys5 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_d @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5094_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_nat,Zs: list_c,P: list_c > list_nat > list_c > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( P @ nil_c @ nil_nat @ nil_c )
         => ( ! [X3: c,Xs2: list_c,Y2: nat,Ys5: list_nat,Z4: c,Zs2: list_c] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_c @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5095_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_nat,Zs: list_d,P: list_c > list_nat > list_d > $o] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_d @ Zs ) )
       => ( ( P @ nil_c @ nil_nat @ nil_d )
         => ( ! [X3: c,Xs2: list_c,Y2: nat,Ys5: list_nat,Z4: d,Zs2: list_d] :
                ( ( ( size_size_list_c @ Xs2 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_d @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys5 @ Zs2 )
                   => ( P @ ( cons_c @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_d @ Z4 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_5096_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_c,P: list_nat > list_nat > list_int > list_c > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_c @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_int @ nil_c )
           => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z4: int,Zs2: list_int,W3: c,Ws2: list_c] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_c @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5097_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_d,P: list_nat > list_nat > list_int > list_d > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_d @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_int @ nil_d )
           => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z4: int,Zs2: list_int,W3: d,Ws2: list_d] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_d @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_d @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5098_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_nat,P: list_nat > list_nat > list_int > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_int @ nil_nat )
           => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z4: int,Zs2: list_int,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5099_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_int,P: list_nat > list_nat > list_int > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_int @ nil_int )
           => ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z4: int,Zs2: list_int,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_int @ Zs2 ) )
                   => ( ( ( size_size_list_int @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5100_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_c,P: list_nat > list_int > list_c > list_c > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( ( size_size_list_c @ Zs )
            = ( size_size_list_c @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_c @ nil_c )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: c,Zs2: list_c,W3: c,Ws2: list_c] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_c @ Zs2 ) )
                   => ( ( ( size_size_list_c @ Zs2 )
                        = ( size_size_list_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) @ ( cons_c @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5101_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_d,P: list_nat > list_int > list_c > list_d > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( ( size_size_list_c @ Zs )
            = ( size_size_list_d @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_c @ nil_d )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: c,Zs2: list_c,W3: d,Ws2: list_d] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_c @ Zs2 ) )
                   => ( ( ( size_size_list_c @ Zs2 )
                        = ( size_size_list_d @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) @ ( cons_d @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5102_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_nat,P: list_nat > list_int > list_c > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( ( size_size_list_c @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_c @ nil_nat )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: c,Zs2: list_c,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_c @ Zs2 ) )
                   => ( ( ( size_size_list_c @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5103_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_int,P: list_nat > list_int > list_c > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_c @ Zs ) )
       => ( ( ( size_size_list_c @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_c @ nil_int )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: c,Zs2: list_c,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_c @ Zs2 ) )
                   => ( ( ( size_size_list_c @ Zs2 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_c @ Z4 @ Zs2 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5104_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_d,Ws: list_c,P: list_nat > list_int > list_d > list_c > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_d @ Zs ) )
       => ( ( ( size_size_list_d @ Zs )
            = ( size_size_list_c @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_d @ nil_c )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: d,Zs2: list_d,W3: c,Ws2: list_c] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_d @ Zs2 ) )
                   => ( ( ( size_size_list_d @ Zs2 )
                        = ( size_size_list_c @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_d @ Z4 @ Zs2 ) @ ( cons_c @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5105_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_d,Ws: list_d,P: list_nat > list_int > list_d > list_d > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_d @ Zs ) )
       => ( ( ( size_size_list_d @ Zs )
            = ( size_size_list_d @ Ws ) )
         => ( ( P @ nil_nat @ nil_int @ nil_d @ nil_d )
           => ( ! [X3: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z4: d,Zs2: list_d,W3: d,Ws2: list_d] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys5 ) )
                 => ( ( ( size_size_list_int @ Ys5 )
                      = ( size_size_list_d @ Zs2 ) )
                   => ( ( ( size_size_list_d @ Zs2 )
                        = ( size_size_list_d @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys5 ) @ ( cons_d @ Z4 @ Zs2 ) @ ( cons_d @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5106_power__gt__expt,axiom,
    ! [N: nat,K3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K3 @ ( power_power_nat @ N @ K3 ) ) ) ).

% power_gt_expt
thf(fact_5107_greaterThan__Suc,axiom,
    ! [K3: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K3 ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K3 ) @ ( insert_nat2 @ ( suc @ K3 ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_5108_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_5109_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_5110_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P = P3 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
           => P3 ) ) ) ) ).

% imp_le_cong
thf(fact_5111_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P = P3 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
            & P3 ) ) ) ) ).

% conj_le_cong
thf(fact_5112_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K3: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
         => ! [X7: int] :
              ( ( P @ X7 )
             => ( P @ ( plus_plus_int @ X7 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_5113_decr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K3: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
         => ! [X7: int] :
              ( ( P @ X7 )
             => ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_5114_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z3 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_5115_plusinfinity,axiom,
    ! [D2: int,P3: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P3 @ X3 )
            = ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z3 @ X3 )
             => ( ( P @ X3 )
                = ( P3 @ X3 ) ) )
         => ( ? [X_12: int] : ( P3 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_5116_pos__int__cases,axiom,
    ! [K3: int] :
      ( ( ord_less_int @ zero_zero_int @ K3 )
     => ~ ! [N5: nat] :
            ( ( K3
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% pos_int_cases
thf(fact_5117_zero__less__imp__eq__int,axiom,
    ! [K3: int] :
      ( ( ord_less_int @ zero_zero_int @ K3 )
     => ? [N5: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N5 )
          & ( K3
            = ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_5118_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K3: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_5119_eucl__rel__int__iff,axiom,
    ! [K3: int,L: int,Q6: int,R2: int] :
      ( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( ( K3
          = ( plus_plus_int @ ( times_times_int @ L @ Q6 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q6 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5120_nat0__intermed__int__val,axiom,
    ! [N: nat,F4: nat > int,K3: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F4 @ ( plus_plus_nat @ I5 @ one_one_nat ) ) @ ( F4 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F4 @ zero_zero_nat ) @ K3 )
       => ( ( ord_less_eq_int @ K3 @ ( F4 @ N ) )
         => ? [I5: nat] :
              ( ( ord_less_eq_nat @ I5 @ N )
              & ( ( F4 @ I5 )
                = K3 ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_5121_unique__remainder,axiom,
    ! [A2: int,B2: int,Q6: int,R2: int,Q7: int,R5: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q6 @ R2 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q7 @ R5 ) )
       => ( R2 = R5 ) ) ) ).

% unique_remainder
thf(fact_5122_unique__quotient,axiom,
    ! [A2: int,B2: int,Q6: int,R2: int,Q7: int,R5: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q6 @ R2 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q7 @ R5 ) )
       => ( Q6 = Q7 ) ) ) ).

% unique_quotient
thf(fact_5123_eucl__rel__int__by0,axiom,
    ! [K3: int] : ( eucl_rel_int @ K3 @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K3 ) ) ).

% eucl_rel_int_by0
thf(fact_5124_eucl__rel__int__dividesI,axiom,
    ! [L: int,K3: int,Q6: int] :
      ( ( L != zero_zero_int )
     => ( ( K3
          = ( times_times_int @ Q6 @ L ) )
       => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q6 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5125_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F4: nat > int,K3: int] :
      ( ! [I5: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I5 )
            & ( ord_less_nat @ I5 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F4 @ ( suc @ I5 ) ) @ ( F4 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F4 @ M2 ) @ K3 )
         => ( ( ord_less_eq_int @ K3 @ ( F4 @ N ) )
           => ? [I5: nat] :
                ( ( ord_less_eq_nat @ M2 @ I5 )
                & ( ord_less_eq_nat @ I5 @ N )
                & ( ( F4 @ I5 )
                  = K3 ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_5126_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_5127_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_5128_nat__ivt__aux,axiom,
    ! [N: nat,F4: nat > int,K3: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F4 @ ( suc @ I5 ) ) @ ( F4 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F4 @ zero_zero_nat ) @ K3 )
       => ( ( ord_less_eq_int @ K3 @ ( F4 @ N ) )
         => ? [I5: nat] :
              ( ( ord_less_eq_nat @ I5 @ N )
              & ( ( F4 @ I5 )
                = K3 ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_5129_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_5130_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K4: int] :
              ( ( A12 = K4 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K4 ) ) )
          | ? [L4: int,K4: int,Q8: int] :
              ( ( A12 = K4 )
              & ( A23 = L4 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ zero_zero_int ) )
              & ( L4 != zero_zero_int )
              & ( K4
                = ( times_times_int @ Q8 @ L4 ) ) )
          | ? [R4: int,L4: int,K4: int,Q8: int] :
              ( ( A12 = K4 )
              & ( A23 = L4 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ R4 ) )
              & ( ( sgn_sgn_int @ R4 )
                = ( sgn_sgn_int @ L4 ) )
              & ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L4 ) )
              & ( K4
                = ( plus_plus_int @ ( times_times_int @ Q8 @ L4 ) @ R4 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5131_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q5: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q5 @ A22 ) ) ) )
         => ~ ! [R6: int,Q5: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q5 @ R6 ) )
               => ( ( ( sgn_sgn_int @ R6 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R6 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A22 ) @ R6 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5132_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K3: int,Q6: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K3
            = ( plus_plus_int @ ( times_times_int @ Q6 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_5133_int__power__div__base,axiom,
    ! [M2: nat,K3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_int @ zero_zero_int @ K3 )
       => ( ( divide_divide_int @ ( power_power_int @ K3 @ M2 ) @ K3 )
          = ( power_power_int @ K3 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_5134_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_5135_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_5136_entt__true,axiom,
    ! [A: assn] : ( entailst @ A @ top_top_assn ) ).

% entt_true
thf(fact_5137_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_5138_entt__star__true__simp_I1_J,axiom,
    ! [A: assn,B: assn] :
      ( ( entailst @ A @ ( times_times_assn @ B @ top_top_assn ) )
      = ( entailst @ A @ B ) ) ).

% entt_star_true_simp(1)
thf(fact_5139_entt__star__true__simp_I2_J,axiom,
    ! [A: assn,B: assn] :
      ( ( entailst @ ( times_times_assn @ A @ top_top_assn ) @ B )
      = ( entailst @ A @ B ) ) ).

% entt_star_true_simp(2)
thf(fact_5140_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_5141_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_5142_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% norm_assertion_simps(12)
thf(fact_5143_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% norm_assertion_simps(11)
thf(fact_5144_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% norm_assertion_simps(4)
thf(fact_5145_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X )
      = X ) ).

% norm_assertion_simps(3)
thf(fact_5146_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_5147_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_5148_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_5149_ent__star__mono__true,axiom,
    ! [A: assn,A6: assn,B: assn,B9: assn] :
      ( ( entails @ A @ ( times_times_assn @ A6 @ top_top_assn ) )
     => ( ( entails @ B @ ( times_times_assn @ B9 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A @ B ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A6 @ B9 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_5150_ent__refl__true,axiom,
    ! [A: assn] : ( entails @ A @ ( times_times_assn @ A @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_5151_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_5152_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_5153_div__int__unique,axiom,
    ! [K3: int,L: int,Q6: int,R2: int] :
      ( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
     => ( ( divide_divide_int @ K3 @ L )
        = Q6 ) ) ).

% div_int_unique
thf(fact_5154_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_5155_entt__def__true,axiom,
    ( entailst
    = ( ^ [P2: assn,Q2: assn] : ( entails @ ( times_times_assn @ P2 @ top_top_assn ) @ ( times_times_assn @ Q2 @ top_top_assn ) ) ) ) ).

% entt_def_true
thf(fact_5156_entailst__def,axiom,
    ( entailst
    = ( ^ [A5: assn,B7: assn] : ( entails @ A5 @ ( times_times_assn @ B7 @ top_top_assn ) ) ) ) ).

% entailst_def
thf(fact_5157_enttI__true,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ top_top_assn ) @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entailst @ P @ Q ) ) ).

% enttI_true
thf(fact_5158_enttI,axiom,
    ! [A: assn,B: assn] :
      ( ( entails @ A @ ( times_times_assn @ B @ top_top_assn ) )
     => ( entailst @ A @ B ) ) ).

% enttI
thf(fact_5159_enttD,axiom,
    ! [A: assn,B: assn] :
      ( ( entailst @ A @ B )
     => ( entails @ A @ ( times_times_assn @ B @ top_top_assn ) ) ) ).

% enttD
thf(fact_5160_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ top_top_assn )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% entails_solve_init(1)
thf(fact_5161_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_5162_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
        = M2 ) ) ).

% div_mult_self_is_m
thf(fact_5163_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
        = M2 ) ) ).

% div_mult_self1_is_m
thf(fact_5164_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M2 @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_5165_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_5166_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ N @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_5167_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K3 @ N ) @ ( divide_divide_nat @ K3 @ M2 ) ) ) ) ).

% div_le_mono2
thf(fact_5168_div__less__iff__less__mult,axiom,
    ! [Q6: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q6 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q6 ) @ N )
        = ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q6 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_5169_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N )
          = M2 )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_5170_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_5171_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_5172_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q6: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q6 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q6 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q6 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_5173_div__nat__eqI,axiom,
    ! [N: nat,Q6: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
       => ( ( divide_divide_nat @ M2 @ N )
          = Q6 ) ) ) ).

% div_nat_eqI
thf(fact_5174_split__div,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J2 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_5175_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_5176_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_5177_nat__mult__div__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
        = ( divide_divide_nat @ M2 @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_5178_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_5179_split__div_H,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q8: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q8 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q8 ) ) )
            & ( P @ Q8 ) ) ) ) ).

% split_div'
thf(fact_5180_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_5181_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = M2 ) ) ).

% mod_less
thf(fact_5182_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_5183_div__less__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A @ N ) @ ( divide_divide_nat @ B @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5184_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P6: nat,M2: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P6 )
       => ( ( ord_less_nat @ M2 @ P6 )
         => ( ! [N5: nat] :
                ( ( ord_less_nat @ N5 @ P6 )
               => ( ( P @ N5 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N5 ) @ P6 ) ) ) )
           => ( P @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_5185_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_5186_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M @ N2 ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_5187_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_5188_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J2 ) )
               => ( P @ J2 ) ) ) ) ) ) ).

% split_mod
thf(fact_5189_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_5190_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_5191_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [M6: nat] : ( P @ M6 @ zero_zero_nat )
     => ( ! [M6: nat,N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ( P @ N5 @ ( modulo_modulo_nat @ M6 @ N5 ) )
             => ( P @ M6 @ N5 ) ) )
       => ( P @ M2 @ N ) ) ) ).

% gcd_nat_induct
thf(fact_5192_mod__int__unique,axiom,
    ! [K3: int,L: int,Q6: int,R2: int] :
      ( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
     => ( ( modulo_modulo_int @ K3 @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5193_eucl__rel__int,axiom,
    ! [K3: int,L: int] : ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K3 @ L ) @ ( modulo_modulo_int @ K3 @ L ) ) ) ).

% eucl_rel_int
thf(fact_5194_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5195_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A3: assn,B3: assn] : ( inf_inf_assn @ A3 @ ( uminus_uminus_assn @ B3 ) ) ) ) ).

% minus_assn_def
thf(fact_5196_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N5: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N5 ) )
     => ~ ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) ) ) ) ).

% int_cases4
thf(fact_5197_int__cases3,axiom,
    ! [K3: int] :
      ( ( K3 != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K3
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
       => ~ ! [N5: nat] :
              ( ( K3
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ) ).

% int_cases3
thf(fact_5198_neg__int__cases,axiom,
    ! [K3: int] :
      ( ( ord_less_int @ K3 @ zero_zero_int )
     => ~ ! [N5: nat] :
            ( ( K3
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% neg_int_cases
thf(fact_5199_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q6: int,R2: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q6 @ R2 ) )
     => ( ( B2 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A2 ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q6 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q6 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5200_entails__solve__finalize_I1_J,axiom,
    ! [M5: list_P8527749157015355191n_assn,P: assn] : ( fI_RESULT @ M5 @ P @ one_one_assn @ top_top_assn ) ).

% entails_solve_finalize(1)
thf(fact_5201_frame__inference__finalize,axiom,
    ! [M5: list_P8527749157015355191n_assn,F: assn] : ( fI_RESULT @ M5 @ F @ one_one_assn @ F ) ).

% frame_inference_finalize
thf(fact_5202_entails__solve__finalize_I2_J,axiom,
    ! [M5: list_P8527749157015355191n_assn] : ( fI_RESULT @ M5 @ one_one_assn @ one_one_assn @ one_one_assn ) ).

% entails_solve_finalize(2)
thf(fact_5203_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_5204_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_5205_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_5206_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_5207_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_5208_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_5209_nth__upto,axiom,
    ! [I: int,K3: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K3 ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K3 )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K3 ) ) ) ) ).

% nth_upto
thf(fact_5210_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_5211_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_5212_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_5213_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_5214_sorted__upto,axiom,
    ! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).

% sorted_upto
thf(fact_5215_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_5216_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I4: int,J2: int] : ( append_int @ ( upto @ I4 @ J2 ) ) ) ) ).

% upto_aux_def
thf(fact_5217_upto__code,axiom,
    ( upto
    = ( ^ [I4: int,J2: int] : ( upto_aux @ I4 @ J2 @ nil_int ) ) ) ).

% upto_code
thf(fact_5218_upto__split2,axiom,
    ! [I: int,J: int,K3: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K3 )
       => ( ( upto @ I @ K3 )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K3 ) ) ) ) ) ).

% upto_split2
thf(fact_5219_upto__split1,axiom,
    ! [I: int,J: int,K3: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K3 )
       => ( ( upto @ I @ K3 )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K3 ) ) ) ) ) ).

% upto_split1
thf(fact_5220_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_5221_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J2 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_5222_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_5223_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_5224_upto__split3,axiom,
    ! [I: int,J: int,K3: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K3 )
       => ( ( upto @ I @ K3 )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K3 ) ) ) ) ) ) ).

% upto_split3
thf(fact_5225_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_5226_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_5227_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_5228_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_5229_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_5230_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_5231_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_5232_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_5233_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_5234_nat__less__iff,axiom,
    ! [W: int,M2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% nat_less_iff
thf(fact_5235_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q6: int,R2: int] :
      ( ( ord_less_eq_int @ B2 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q6 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q6 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5236_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q6: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q6 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q6 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5237_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_5238_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_5239_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_5240_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_5241_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_5242_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_5243_ex__power__ivl1,axiom,
    ! [B2: nat,K3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K3 )
       => ? [N5: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N5 ) @ K3 )
            & ( ord_less_nat @ K3 @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_5244_ex__power__ivl2,axiom,
    ! [B2: nat,K3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 )
       => ? [N5: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N5 ) @ K3 )
            & ( ord_less_eq_nat @ K3 @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_5245_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( P @ N5 )
           => ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) )
       => ( ! [N5: nat] :
              ( ( P @ N5 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_5246_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_5247_uminus__dvd__conv_I1_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D5: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D5 ) ) ) ) ).

% uminus_dvd_conv(1)
thf(fact_5248_uminus__dvd__conv_I2_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D5: int,T2: int] : ( dvd_dvd_int @ D5 @ ( uminus_uminus_int @ T2 ) ) ) ) ).

% uminus_dvd_conv(2)
thf(fact_5249_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% nat_dvd_not_less
thf(fact_5250_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
       => ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).

% dvd_pos_nat
thf(fact_5251_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_minus_self
thf(fact_5252_zdvd__mono,axiom,
    ! [K3: int,M2: int,T: int] :
      ( ( K3 != zero_zero_int )
     => ( ( dvd_dvd_int @ M2 @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K3 @ M2 ) @ ( times_times_int @ K3 @ T ) ) ) ) ).

% zdvd_mono
thf(fact_5253_dvd__imp__le,axiom,
    ! [K3: nat,N: nat] :
      ( ( dvd_dvd_nat @ K3 @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K3 @ N ) ) ) ).

% dvd_imp_le
thf(fact_5254_nat__mult__dvd__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
        = ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_5255_dvd__mult__cancel,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_5256_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_5257_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_5258_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_5259_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_5260_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M2 )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
         => ( ( modulo_modulo_nat @ M2 @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_5261_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_5262_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 ) ) ).

% take_bit_nat_eq_self
thf(fact_5263_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_5264_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 )
      = ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_5265_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).

% take_bit_nat_less_self_iff
thf(fact_5266_take__bit__int__less__eq,axiom,
    ! [N: nat,K3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( minus_minus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_5267_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_5268_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% even_diff_nat
thf(fact_5269_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_5270_Divides_Oadjust__div__eq,axiom,
    ! [Q6: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5271_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_5272_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N5: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N5 ) ) )
       => ( ! [N5: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N5 ) ) )
         => ( ! [M6: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M6 ) @ one ) )
           => ( ! [M6: num,N5: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M6 ) @ ( bit0 @ N5 ) ) )
             => ( ! [M6: num,N5: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M6 ) @ ( bit1 @ N5 ) ) )
               => ( ! [M6: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M6 ) @ one ) )
                 => ( ! [M6: num,N5: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M6 ) @ ( bit0 @ N5 ) ) )
                   => ~ ! [M6: num,N5: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M6 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_5273_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5274_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_5275_divmod__BitM__2__eq,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_5276_less__numeral__Suc,axiom,
    ! [K3: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K3 ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K3 ) @ N ) ) ).

% less_numeral_Suc
thf(fact_5277_less__Suc__numeral,axiom,
    ! [N: nat,K3: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K3 ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K3 ) ) ) ).

% less_Suc_numeral
thf(fact_5278_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M @ N2 ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_5279_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_5280_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_5281_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_5282_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( 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 @ X @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_5283_and__int_Opsimps,axiom,
    ! [K3: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L ) )
     => ( ( ( ( member_int @ K3 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K3 @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K3 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K3 @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_5284_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K3: int] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( bit_se1146084159140164899it_int @ K3 @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K3 ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_5285_int__bit__bound,axiom,
    ! [K3: int] :
      ~ ! [N5: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N5 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K3 @ M3 )
                = ( bit_se1146084159140164899it_int @ K3 @ N5 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( bit_se1146084159140164899it_int @ K3 @ ( minus_minus_nat @ N5 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K3 @ N5 ) ) ) ) ) ).

% int_bit_bound
thf(fact_5286_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 ) )
     => ( ! [K: int,L2: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
           => ( ( ~ ( ( member_int @ K @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K @ L2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_5287_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_5288_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_5289_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I5: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I5 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I5 @ J3 )
               => ( P @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) )
             => ( P @ I5 @ J3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_5290_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_5291_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K4: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K4 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K4 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K4 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ) ) ).

% binomial_code
thf(fact_5292_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_nat2 @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_5293_binomial__eq__0__iff,axiom,
    ! [N: nat,K3: nat] :
      ( ( ( binomial @ N @ K3 )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K3 ) ) ).

% binomial_eq_0_iff
thf(fact_5294_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_5295_zero__less__binomial__iff,axiom,
    ! [N: nat,K3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K3 ) )
      = ( ord_less_eq_nat @ K3 @ N ) ) ).

% zero_less_binomial_iff
thf(fact_5296_binomial__eq__0,axiom,
    ! [N: nat,K3: nat] :
      ( ( ord_less_nat @ N @ K3 )
     => ( ( binomial @ N @ K3 )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_5297_zero__less__binomial,axiom,
    ! [K3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K3 @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K3 ) ) ) ).

% zero_less_binomial
thf(fact_5298_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M6: nat] :
          ( X
         != ( product_Pair_nat_nat @ K @ M6 ) ) ).

% prod_decode_aux.cases
thf(fact_5299_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_5300_choose__reduce__nat,axiom,
    ! [N: nat,K3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K3 )
       => ( ( binomial @ N @ K3 )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K3 ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_5301_times__binomial__minus1__eq,axiom,
    ! [K3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K3 )
     => ( ( times_times_nat @ K3 @ ( binomial @ N @ K3 ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_5302_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_5303_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_5304_binomial__addition__formula,axiom,
    ! [N: nat,K3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K3 ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K3 ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K3 ) ) ) ) ).

% binomial_addition_formula
thf(fact_5305_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_5306_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K4: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K4 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_5307_cppi,axiom,
    ! [D: int,P: int > $o,P3: int > $o,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( P @ X3 )
              = ( P3 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P3 @ X3 )
                = ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P3 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ A )
                        & ( P @ ( minus_minus_int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_5308_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ I4 @ J2 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_5309_bset_I1_J,axiom,
    ! [D: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D ) ) ) )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
             => ( ( P @ ( minus_minus_int @ X7 @ D ) )
                & ( Q @ ( minus_minus_int @ X7 @ D ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_5310_bset_I2_J,axiom,
    ! [D: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D ) ) ) )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
             => ( ( P @ ( minus_minus_int @ X7 @ D ) )
                | ( Q @ ( minus_minus_int @ X7 @ D ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_5311_aset_I1_J,axiom,
    ! [D: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D ) ) ) )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
             => ( ( P @ ( plus_plus_int @ X7 @ D ) )
                & ( Q @ ( plus_plus_int @ X7 @ D ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_5312_aset_I2_J,axiom,
    ! [D: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D ) ) ) )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
             => ( ( P @ ( plus_plus_int @ X7 @ D ) )
                | ( Q @ ( plus_plus_int @ X7 @ D ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_5313_aset_I10_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_5314_aset_I9_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_5315_bset_I10_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_5316_bset_I9_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_5317_periodic__finite__ex,axiom,
    ! [D2: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_5318_aset_I7_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X7 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X7 @ D ) ) ) ) ) ).

% aset(7)
thf(fact_5319_aset_I5_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ A )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X7 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_5320_aset_I4_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ A )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 != T )
             => ( ( plus_plus_int @ X7 @ D )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_5321_aset_I3_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 = T )
             => ( ( plus_plus_int @ X7 @ D )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_5322_bset_I7_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ B )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X7 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X7 @ D ) ) ) ) ) ) ).

% bset(7)
thf(fact_5323_bset_I5_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X7 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X7 @ D ) @ T ) ) ) ) ).

% bset(5)
thf(fact_5324_bset_I4_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ B )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 != T )
             => ( ( minus_minus_int @ X7 @ D )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_5325_bset_I3_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 = T )
             => ( ( minus_minus_int @ X7 @ D )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_5326_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I4 ) @ bot_bot_set_int @ ( insert_int2 @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) ) ) ) ).

% simp_from_to
thf(fact_5327_bset_I6_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X7 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X7 @ D ) @ T ) ) ) ) ).

% bset(6)
thf(fact_5328_bset_I8_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X7 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X7 @ D ) ) ) ) ) ) ).

% bset(8)
thf(fact_5329_aset_I6_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X7 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X7 @ D ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_5330_aset_I8_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X7 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X7 @ D ) ) ) ) ) ).

% aset(8)
thf(fact_5331_cpmi,axiom,
    ! [D: int,P: int > $o,P3: int > $o,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( P @ X3 )
              = ( P3 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P3 @ X3 )
                = ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P3 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ B )
                        & ( P @ ( plus_plus_int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_5332_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% less_eq
thf(fact_5333_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M: nat] :
            ( ( ord_less_eq_nat @ M @ N )
           => ( P @ M ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_5334_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M: nat] :
            ( ( ord_less_eq_nat @ M @ N )
            & ( P @ M ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_5335_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat2 @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_5336_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( insert_nat2 @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5337_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
        = ( insert_nat2 @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5338_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N )
        = ( insert_nat2 @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5339_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_5340_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_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_5341_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_5342_atMost__Suc,axiom,
    ! [K3: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K3 ) )
      = ( insert_nat2 @ ( suc @ K3 ) @ ( set_ord_atMost_nat @ K3 ) ) ) ).

% atMost_Suc
thf(fact_5343_atMost__nat__numeral,axiom,
    ! [K3: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K3 ) )
      = ( insert_nat2 @ ( numeral_numeral_nat @ K3 ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K3 ) ) ) ) ).

% atMost_nat_numeral
thf(fact_5344_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_5345_sorted__list__of__set__atMost__Suc,axiom,
    ! [K3: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K3 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K3 ) ) @ ( cons_nat @ ( suc @ K3 ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_5346_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_5347_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5348_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_5349_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5350_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N8 )
         => ( ord_less_nat @ X2 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_5351_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_5352_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_5353_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( gcd_Gcd_nat @ M5 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_5354_set__encode__insert,axiom,
    ! [A: set_nat,N: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ N @ A )
       => ( ( nat_set_encode @ ( insert_nat2 @ N @ A ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A ) ) ) ) ) ).

% set_encode_insert
thf(fact_5355_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_5356_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_5357_unbounded__k__infinite,axiom,
    ! [K3: nat,S: set_nat] :
      ( ! [M6: nat] :
          ( ( ord_less_nat @ K3 @ M6 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M6 @ N9 )
              & ( member_nat @ N9 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_5358_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_5359_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K3: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K3 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K3 ) ) @ ( cons_nat @ K3 @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_5360_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_5361_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat2 @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_5362_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M: nat] :
            ( ( ord_less_nat @ M @ N )
            & ( P @ M ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_5363_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M: nat] :
            ( ( ord_less_nat @ M @ N )
           => ( P @ M ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_5364_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_5365_lessThan__Suc,axiom,
    ! [K3: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K3 ) )
      = ( insert_nat2 @ K3 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% lessThan_Suc
thf(fact_5366_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_5367_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_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_5368_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat2 @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_5369_lessThan__nat__numeral,axiom,
    ! [K3: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K3 ) )
      = ( insert_nat2 @ ( pred_numeral @ K3 ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K3 ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_5370_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_5371_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = ( insert_nat2 @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_5372_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5373_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K3: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K3 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K3 ) )
          = ( insert_nat2 @ ( pred_numeral @ K3 ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K3 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K3 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K3 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_5374_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_5375_atLeast__Suc,axiom,
    ! [K3: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K3 ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K3 ) @ ( insert_nat2 @ K3 @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_5376_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_5377_finite__Collect__less__nat,axiom,
    ! [K3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K3 ) ) ) ).

% finite_Collect_less_nat
thf(fact_5378_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ ( suc @ K4 ) @ M5 )
                & ( ord_less_nat @ K4 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M5 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_5379_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K4: nat] :
                  ( ( member_nat @ ( suc @ K4 ) @ M5 )
                  & ( ord_less_nat @ K4 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M5 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_5380_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M5 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_5381_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K4: nat] :
            ( ( P @ K4 )
            & ( ord_less_nat @ K4 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_5382_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_less_as_int
thf(fact_5383_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) ) ) ) ).

% finite_divisors_nat
thf(fact_5384_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M: nat] : ( modulo_modulo_nat @ M @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_5385_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_5386_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_5387_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_5388_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q8: nat] : ( product_Pair_nat_nat @ ( suc @ Q8 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_5389_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L4: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q8: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L4 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L4 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_5390_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L4: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q8: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L4 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L4 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ R4 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_5391_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_5392_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y5 @ N )
              & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_5393_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] :
                ( ( ord_less_nat @ X2 @ N )
                & ( ord_less_nat @ Y5 @ N )
                & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_5394_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_5395_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X2: int] : X2
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_5396_remdups__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% remdups_upt
thf(fact_5397_sort__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X2: nat] : X2
        @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sort_upt
thf(fact_5398_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_5399_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_5400_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_5401_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_5402_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_5403_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_5404_upt__merge,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ J @ K3 ) )
     => ( ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ K3 ) )
        = ( upt @ I @ K3 ) ) ) ).

% upt_merge
thf(fact_5405_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sorted_list_of_set_range
thf(fact_5406_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_5407_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_5408_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_5409_nth__upt,axiom,
    ! [I: nat,K3: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K3 )
        = ( plus_plus_nat @ I @ K3 ) ) ) ).

% nth_upt
thf(fact_5410_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_5411_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I4: nat,J2: nat] : ( set_nat2 @ ( upt @ I4 @ J2 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_5412_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_5413_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
        @ ( upt @ A2 @ B2 ) )
      = ( upt @ ( plus_plus_nat @ A2 @ Ofs ) @ ( plus_plus_nat @ B2 @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_5414_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_5415_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N2: nat,M: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_5416_butlast__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( butlast_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_5417_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_5418_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q6: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q6 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q6 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_5419_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_5420_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).

% sorted_wrt_upt
thf(fact_5421_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_5422_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_5423_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_5424_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs @ Ys ) )
        = ( ? [K4: nat] :
              ( ( ord_less_eq_nat @ I @ K4 )
              & ( ord_less_eq_nat @ K4 @ J )
              & ( ( upt @ I @ K4 )
                = Xs )
              & ( ( upt @ K4 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_5425_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_5426_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_5427_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_5428_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I5: nat] :
            ( ( ( ord_less_eq_nat @ U @ I5 )
              & ( ord_less_nat @ I5 @ U3 ) )
           => ~ ( P @ I5 ) )
       => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_5429_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_5430_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_5431_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K3 ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K3 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_5432_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_5433_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J2 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_5434_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_5435_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_5436_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_5437_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L4: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q8: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L4 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L4 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ R4 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_5438_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_5439_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5440_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_5441_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_5442_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_5443_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [R6: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R6 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
             => ( member_nat @ ( R6 @ N9 ) @ S ) ) ) ) ).

% finite_enumerate
thf(fact_5444_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_5445_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_5446_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_5447_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_5448_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_5449_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K4 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_5450_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L4: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K4 @ L4 ) @ ( modulo364778990260209775nteger @ K4 @ L4 ) ) ) ) ).

% divmod_integer_def
thf(fact_5451_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R4: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R4 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_5452_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
     => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
        = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_5453_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_5454_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_5455_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
          = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_5456_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K4: code_integer,L4: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L4 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L4 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_5457_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L4: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L4 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ ( code_divmod_abs @ K4 @ L4 )
              @ ( produc6916734918728496179nteger
                @ ^ [R4: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L4 @ S6 ) ) )
                @ ( code_divmod_abs @ K4 @ L4 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K4 @ L4 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R4: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L4 ) @ S6 ) ) )
                    @ ( code_divmod_abs @ K4 @ L4 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_5458_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ( gcd_Gcd_nat @ M5 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) )
                  @ M5 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_5459_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_5460_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U4 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y5 @ U4 ) ) ) )
          @ Xa
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_5461_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_5462_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y2: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) ) ).

% eq_Abs_Integ
thf(fact_5463_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_5464_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_5465_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_5466_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_5467_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U4 @ Y5 ) ) )
        @ Xa
        @ X ) ) ).

% less_int.abs_eq
thf(fact_5468_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U4 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
          @ Xa
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_5469_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y5 @ U4 ) ) )
          @ Xa
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_5470_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L4: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L4
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K4 )
                  = ( sgn_sgn_Code_integer @ L4 ) )
                @ ( code_divmod_abs @ K4 @ L4 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R4: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L4 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K4 @ L4 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_5471_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y5: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U4 @ Z5 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_5472_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A2: assn,B2: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A2 )
     => ( ( times_times_assn @ A2 @ B2 )
        = ( times_times_assn @ B2 @ A2 ) ) ) ).

% assn_aci(11)
thf(fact_5473_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) ) ) ) ).

% uminus_int_def
thf(fact_5474_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U4 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y5 @ U4 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_5475_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y5 @ U4 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_5476_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y5: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U4 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_5477_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
    @ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_5478_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_5479_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U4 @ Y5 ) ) ) ) ) ) ).

% less_int_def
thf(fact_5480_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_5481_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X2 @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X2 @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X2 @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_5482_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_5483_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_5484_mod__pure,axiom,
    ! [B2: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ H2 )
      = ( ( ( produc8586169260539613262et_nat @ H2 )
          = bot_bot_set_nat )
        & B2 ) ) ).

% mod_pure
thf(fact_5485_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H2 )
      = ( ( produc8586169260539613262et_nat @ H2 )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_5486_fi__match__entails,axiom,
    ! [M2: list_P8527749157015355191n_assn] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ M2 ) )
         => ( produc7274209992780475162assn_o @ entails @ X3 ) )
     => ( entails @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M2 ) @ one_one_assn ) @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M2 ) @ one_one_assn ) ) ) ).

% fi_match_entails
thf(fact_5487_mod__star__trueE_H,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H5 )
              = ( produc1824681642469235216et_nat @ H2 ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H5 ) @ ( produc8586169260539613262et_nat @ H2 ) )
             => ~ ( rep_assn @ P @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_5488_FI__RESULT__def,axiom,
    ( fI_RESULT
    = ( ^ [M7: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F3: assn] :
          ( ! [X2: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ M7 ) )
             => ( produc7274209992780475162assn_o @ entails @ X2 ) )
         => ( entails @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M7 ) @ one_one_assn ) @ UP ) @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M7 ) @ one_one_assn ) @ UQ ) @ F3 ) ) ) ) ) ).

% FI_RESULT_def
thf(fact_5489_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_5490_FI__def,axiom,
    ( fi
    = ( ^ [M: list_P8527749157015355191n_assn,P7: assn,Q8: assn,Up: assn,Uq: assn,F6: assn] :
          ( ! [X2: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ M ) )
             => ( produc7274209992780475162assn_o @ entails @ X2 ) )
         => ( entails @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M ) @ one_one_assn ) @ P7 ) @ Up ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M ) @ one_one_assn ) @ Q8 ) @ Uq ) @ F6 ) ) ) ) ) ).

% FI_def
thf(fact_5491_FI__p__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Ps2: assn,Qs: assn,Q6: assn,P6: assn,Up2: assn,Uq2: assn,F4: assn] :
      ( ( fi @ M2 @ Ps2 @ ( times_times_assn @ Qs @ Q6 ) @ ( times_times_assn @ P6 @ Up2 ) @ Uq2 @ F4 )
     => ( fi @ M2 @ ( times_times_assn @ Ps2 @ P6 ) @ ( times_times_assn @ Qs @ Q6 ) @ Up2 @ Uq2 @ F4 ) ) ).

% FI_p_nomatch
thf(fact_5492_FI__finalize,axiom,
    ! [M2: list_P8527749157015355191n_assn,P6: assn,Up2: assn,Q6: assn,Uq2: assn,F4: assn] :
      ( ( fI_RESULT @ M2 @ ( times_times_assn @ P6 @ Up2 ) @ ( times_times_assn @ Q6 @ Uq2 ) @ F4 )
     => ( fi @ M2 @ P6 @ Q6 @ Up2 @ Uq2 @ F4 ) ) ).

% FI_finalize
thf(fact_5493_FI__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( ( fi @ nil_Pr5671120429643327159n_assn @ ( times_times_assn @ sln @ P ) @ ( times_times_assn @ sln @ Q ) @ sln @ sln @ F )
     => ( fI_QUERY @ P @ Q @ F ) ) ).

% FI_init
thf(fact_5494_SLN__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ sln )
      = P ) ).

% SLN_right
thf(fact_5495_SLN__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ sln @ P )
      = P ) ).

% SLN_left
thf(fact_5496_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
thf(fact_5497_FI__q__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Up2: assn,Qs: assn,Q6: assn,Uq2: assn,F4: assn] :
      ( ( fi @ M2 @ ( times_times_assn @ sln @ Up2 ) @ Qs @ sln @ ( times_times_assn @ Q6 @ Uq2 ) @ F4 )
     => ( fi @ M2 @ sln @ ( times_times_assn @ Qs @ Q6 ) @ Up2 @ Uq2 @ F4 ) ) ).

% FI_q_nomatch
thf(fact_5498_FI__match,axiom,
    ! [P6: assn,Q6: assn,M2: list_P8527749157015355191n_assn,Ps2: assn,Up2: assn,Qs: assn,Uq2: assn,F4: assn] :
      ( ( entails @ P6 @ Q6 )
     => ( ( fi @ ( cons_P2971678138204555879n_assn @ ( produc118845697133431529n_assn @ P6 @ Q6 ) @ M2 ) @ ( times_times_assn @ Ps2 @ Up2 ) @ ( times_times_assn @ Qs @ Uq2 ) @ sln @ sln @ F4 )
       => ( fi @ M2 @ ( times_times_assn @ Ps2 @ P6 ) @ ( times_times_assn @ Qs @ Q6 ) @ Up2 @ Uq2 @ F4 ) ) ) ).

% FI_match
thf(fact_5499_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_5500_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat2 @ ( suc @ N ) @ A ) )
      = ( insert_nat2 @ N @ ( vimage_nat_nat @ suc @ A ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_5501_vimage__Suc__insert__0,axiom,
    ! [A: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat2 @ zero_zero_nat @ A ) )
      = ( vimage_nat_nat @ suc @ A ) ) ).

% vimage_Suc_insert_0
thf(fact_5502_finite__le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_5503_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_5504_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U4 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y5 @ U4 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_5505_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_5506_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y4: nat,Z2: nat] : Y4 = Z2
    @ pcr_int
    @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_5507_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_5508_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_5509_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U4 @ Y5 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_5510_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U4 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_5511_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y5 @ U4 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_5512_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U4 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y5 @ U4 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U4 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y5 @ U4 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_5513_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V2 ) )
      = ( ( plus_plus_nat @ X @ V2 )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_5514_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_5515_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_5516_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_atMost_at_top
thf(fact_5517_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_lessThan_at_top
thf(fact_5518_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) ) ) ).

% uminus_int.rsp
thf(fact_5519_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_5520_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y4: $o,Z2: $o] : Y4 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U4 @ Y5 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y5: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U4: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U4 @ Y5 ) ) ) ) ) ).

% less_int.rsp
thf(fact_5521_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y5 @ U4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y5 @ U4 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_5522_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U4 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y5: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U4: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U4 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_5523_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_5524_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_5525_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K3: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_5526_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N2: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_5527_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_5528_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N8 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_5529_le__sequentially,axiom,
    ! [F: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F ) ) ) ).

% le_sequentially
thf(fact_5530_bit__concat__bit__iff,axiom,
    ! [M2: nat,K3: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K3 @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M2 )
          & ( bit_se1146084159140164899it_int @ K3 @ N ) )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_5531_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X2 )
    @ ^ [X2: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X2 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_5532_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_5533_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_5534_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_5535_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_5536_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_5537_smin__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] :
      ( ( X4 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_5538_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_5539_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_5540_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_5541_wmax__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X4 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X4 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_5542_wmin__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_5543_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_5544_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_5545_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_5546_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y6 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y6 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X4 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_5547_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_5548_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y6 )
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_5549_wmsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat] :
      ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
        | ( ( A = zero_z1048942125864253310at_nat )
          & ( B = zero_z1048942125864253310at_nat ) ) )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z6 @ A ) @ ( plus_p7104986032573967614at_nat @ Z6 @ B ) ) @ ms_weak ) ) ).

% wmsI
thf(fact_5550_smsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z6 @ A ) @ ( plus_p7104986032573967614at_nat @ Z6 @ B ) ) @ ms_strict ) ) ).

% smsI
thf(fact_5551_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat @ ( produc5245064249948416855at_nat @ ms_strict @ ms_weak ) ).

% ms_reduction_pair
thf(fact_5552_ms__weakI1,axiom,
    ! [Z6: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z6 @ Z7 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z6 @ A ) @ ( plus_p7104986032573967614at_nat @ Z7 @ B ) ) @ ms_weak ) ) ) ).

% ms_weakI1
thf(fact_5553_ms__strictI,axiom,
    ! [Z6: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z6 @ Z7 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z6 @ A ) @ ( plus_p7104986032573967614at_nat @ Z7 @ B ) ) @ ms_strict ) ) ) ).

% ms_strictI
thf(fact_5554_pw__leq__lstep,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( pw_leq @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) ) ) ).

% pw_leq_lstep
thf(fact_5555_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X4: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( ( pw_leq @ X4 @ Y6 )
       => ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X4 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y6 ) ) ) ) ).

% pw_leq_step
thf(fact_5556_pw__leq_Osimps,axiom,
    ( pw_leq
    = ( ^ [A12: multis2468970476368604999at_nat,A23: multis2468970476368604999at_nat] :
          ( ( ( A12 = zero_z1048942125864253310at_nat )
            & ( A23 = zero_z1048942125864253310at_nat ) )
          | ? [X2: product_prod_nat_nat,Y5: product_prod_nat_nat,X8: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
              ( ( A12
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X2 @ zero_z1048942125864253310at_nat ) @ X8 ) )
              & ( A23
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y5 @ zero_z1048942125864253310at_nat ) @ Y7 ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ fun_pair_leq )
              & ( pw_leq @ X8 @ Y7 ) ) ) ) ) ).

% pw_leq.simps
thf(fact_5557_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( ( pw_leq @ A1 @ A22 )
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A22 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X3: product_prod_nat_nat,Y2: product_prod_nat_nat,X9: multis2468970476368604999at_nat] :
              ( ( A1
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X3 @ zero_z1048942125864253310at_nat ) @ X9 ) )
             => ! [Y8: multis2468970476368604999at_nat] :
                  ( ( A22
                    = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y2 @ zero_z1048942125864253310at_nat ) @ Y8 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y2 ) @ fun_pair_leq )
                   => ~ ( pw_leq @ X9 @ Y8 ) ) ) ) ) ) ).

% pw_leq.cases
thf(fact_5558_ms__weakI2,axiom,
    ! [Z6: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z6 @ Z7 )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z6 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z7 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).

% ms_weakI2
thf(fact_5559_pw__leq__split,axiom,
    ! [X4: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( ( pw_leq @ X4 @ Y6 )
     => ? [A9: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat] :
          ( ( X4
            = ( plus_p7104986032573967614at_nat @ A9 @ Z8 ) )
          & ( Y6
            = ( plus_p7104986032573967614at_nat @ B5 @ Z8 ) )
          & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A9 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
            | ( ( B5 = zero_z1048942125864253310at_nat )
              & ( A9 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
thf(fact_5560_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_5561_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y5 @ N )
              & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_5562_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] :
                ( ( ord_less_nat @ X2 @ N )
                & ( ord_less_nat @ Y5 @ N )
                & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y5 @ N )
              & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_5563_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N8: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ N8 @ M )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N8 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_5564_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o2 @ $false @ ( insert_o2 @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_5565_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_5566_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_5567_natLeq__underS__less,axiom,
    ! [N: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).

% natLeq_underS_less
thf(fact_5568_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu2: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y5 @ N )
              & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_5569_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) )
          @ ^ [Uu2: nat] :
              ( collect_nat
              @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y5 @ N )
              & ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_5570_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_5571_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_5572_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_5573_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_5574_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_5575_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y5 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_5576_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_5577_normalize__negative,axiom,
    ! [Q6: int,P6: int] :
      ( ( ord_less_int @ Q6 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q6 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q6 ) ) ) ) ) ).

% normalize_negative
thf(fact_5578_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_5579_normalize__denom__zero,axiom,
    ! [P6: int] :
      ( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_5580_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P6: int,Q6: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% normalize_denom_pos
thf(fact_5581_normalize__crossproduct,axiom,
    ! [Q6: int,S3: int,P6: int,R2: int] :
      ( ( Q6 != zero_zero_int )
     => ( ( S3 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q6 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S3 ) ) )
         => ( ( times_times_int @ P6 @ S3 )
            = ( times_times_int @ R2 @ Q6 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_5582_rat__inverse__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_inverse_code
thf(fact_5583_normalize__def,axiom,
    ( normalize
    = ( ^ [P7: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P7 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P7 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P7 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P7 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P7 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P7 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P7 ) @ ( product_snd_int_int @ P7 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_5584_quotient__of__number_I3_J,axiom,
    ! [K3: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K3 ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_5585_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_5586_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_5587_quotient__of__number_I5_J,axiom,
    ! [K3: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K3 ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_5588_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_5589_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] :
      ( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
      = ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).

% Gcd_int_set_eq_fold
thf(fact_5590_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N @ D2 ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_5591_quotient__of__denom__pos,axiom,
    ! [R2: rat,P6: int,Q6: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% quotient_of_denom_pos
thf(fact_5592_rat__abs__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_abs_code
thf(fact_5593_rat__sgn__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P6 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P6 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_5594_rat__uminus__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_uminus_code
thf(fact_5595_rat__divide__code,axiom,
    ! [P6: rat,Q6: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C5 @ B3 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_divide_code
thf(fact_5596_rat__minus__code,axiom,
    ! [P6: rat,Q6: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C5 ) ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_minus_code
thf(fact_5597_rat__times__code,axiom,
    ! [P6: rat,Q6: rat] :
      ( ( quotient_of @ ( times_times_rat @ P6 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_times_code
thf(fact_5598_rat__plus__code,axiom,
    ! [P6: rat,Q6: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P6 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C5 ) ) @ ( times_times_int @ C5 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_plus_code
thf(fact_5599_quotient__of__int,axiom,
    ! [A2: int] :
      ( ( quotient_of @ ( of_int @ A2 ) )
      = ( product_Pair_int_int @ A2 @ one_one_int ) ) ).

% quotient_of_int
thf(fact_5600_Frct__code__post_I5_J,axiom,
    ! [K3: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K3 ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K3 ) ) ) ).

% Frct_code_post(5)
thf(fact_5601_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_5602_Gcd__in,axiom,
    ! [A: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member_nat @ A4 @ A )
         => ( ( member_nat @ B4 @ A )
           => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A ) ) )
     => ( ( A != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A ) @ A ) ) ) ).

% Gcd_in
thf(fact_5603_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_5604_Frct__code__post_I6_J,axiom,
    ! [K3: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K3 ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K3 ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_5605_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M2 @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M2 )
                & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_5606_Frct__code__post_I4_J,axiom,
    ! [K3: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K3 ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K3 ) ) ).

% Frct_code_post(4)
thf(fact_5607_Frct__code__post_I2_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ A2 @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_5608_Frct__code__post_I1_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A2 ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_5609_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_5610_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ A2 @ ( uminus_uminus_int @ B2 ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(8)
thf(fact_5611_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(7)
thf(fact_5612_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_5613_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ).

% plus_rat.rsp
thf(fact_5614_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_5615_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_5616_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y4: int,Z2: int] : Y4 = Z2
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y4: int,Z2: int] : Y4 = Z2
      @ ratrel )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).

% Fract.rsp
thf(fact_5617_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y5 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) )
    @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y5 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ).

% times_rat.rsp
thf(fact_5618_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ).

% uminus_rat.rsp
thf(fact_5619_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_5620_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X @ X )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_5621_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_5622_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_5623_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_5624_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% uminus_rat.abs_eq
thf(fact_5625_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_5626_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X2 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_5627_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ) ).

% uminus_rat_def
thf(fact_5628_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ) ).

% plus_rat_def
thf(fact_5629_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y5: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y5 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y5 ) ) ) ) ) ).

% times_rat_def
thf(fact_5630_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_5631_Abs__assn__eqI_I2_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( P @ H )
          = ( rep_assn @ Pr @ H ) )
     => ( Pr
        = ( abs_assn @ P ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_5632_Abs__assn__eqI_I1_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( P @ H )
          = ( rep_assn @ Pr @ H ) )
     => ( ( abs_assn @ P )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_5633_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B3: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B3 ) ) ) ) ).

% pure_assn_def
thf(fact_5634_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_5635_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_5636_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P2: assn,Q2: assn] :
          ( abs_assn
          @ ^ [H3: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P2 @ H3 )
              | ( rep_assn @ Q2 @ H3 ) ) ) ) ) ).

% sup_assn_def
thf(fact_5637_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P2: assn,Q2: assn] :
          ( abs_assn
          @ ^ [H3: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P2 @ H3 )
              & ( rep_assn @ Q2 @ H3 ) ) ) ) ) ).

% inf_assn_def
thf(fact_5638_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P2: assn,Q2: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).

% times_assn_def
thf(fact_5639_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P2: assn,Q2: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).

% wand_assn_def
thf(fact_5640_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P2: assn] :
          ( abs_assn
          @ ^ [H3: produc3658429121746597890et_nat] :
              ( ( in_range @ H3 )
              & ~ ( rep_assn @ P2 @ H3 ) ) ) ) ) ).

% uminus_assn_def
thf(fact_5641_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_5642_mod__true,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H2 )
      = ( in_range @ H2 ) ) ).

% mod_true
thf(fact_5643_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,As3: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As2 @ As3 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) ) ) ) ).

% in_range_dist_union
thf(fact_5644_mod__not__dist,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P ) @ H2 )
      = ( ( in_range @ H2 )
        & ~ ( rep_assn @ P @ H2 ) ) ) ).

% mod_not_dist
thf(fact_5645_in__range__subset,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As2 @ As3 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% in_range_subset
thf(fact_5646_models__in__range,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( in_range @ H2 ) ) ).

% models_in_range
thf(fact_5647_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_5648_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,R: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( ( inf_inf_set_nat @ As2 @ As4 )
              = bot_bot_set_nat )
           => ( ( relH @ As2 @ H2 @ H5 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
               => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
                 => ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q @ R ) @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% wand_assnI
thf(fact_5649_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
              & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As @ As4 )
                      = bot_bot_set_nat )
                    & ( relH @ As @ H @ H5 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H5 @ As4 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_5650_relH__dist__union,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As2 @ As3 ) @ H2 @ H4 )
      = ( ( relH @ As2 @ H2 @ H4 )
        & ( relH @ As3 @ H2 @ H4 ) ) ) ).

% relH_dist_union
thf(fact_5651_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( relH @ As2 @ H2 @ H2 ) ) ).

% relH_refl
thf(fact_5652_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H4 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ).

% relH_in_rangeI(1)
thf(fact_5653_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H4 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ).

% relH_in_rangeI(2)
thf(fact_5654_mod__relH,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,P: assn] :
      ( ( relH @ As2 @ H2 @ H4 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ) ).

% mod_relH
thf(fact_5655_relH__trans,axiom,
    ! [As2: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H12 @ H23 )
     => ( ( relH @ As2 @ H23 @ H32 )
       => ( relH @ As2 @ H12 @ H32 ) ) ) ).

% relH_trans
thf(fact_5656_relH__sym,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H4 )
     => ( relH @ As2 @ H4 @ H2 ) ) ).

% relH_sym
thf(fact_5657_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( relH @ Bs @ H2 @ H4 )
     => ( ( ord_less_eq_set_nat @ As2 @ Bs )
       => ( relH @ As2 @ H2 @ H4 ) ) ) ).

% relH_subset
thf(fact_5658_wand__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( wand_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As2 @ As5 )
                = bot_bot_set_nat )
              & ( relH @ As2 @ H2 @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
              & ( P @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
           => ( Q @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As5 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_5659_wand__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As5 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_5660_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
                & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As @ As6 )
                        = bot_bot_set_nat )
                      & ( relH @ As @ H @ H7 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_5661_wand__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
                  & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As @ As4 )
                          = bot_bot_set_nat )
                        & ( relH @ As @ H @ H5 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H5 @ As4 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_5662_wand__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_5663_wand__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H @ As ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As5 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_5664_normalize__stable,axiom,
    ! [Q6: int,P6: int] :
      ( ( ord_less_int @ zero_zero_int @ Q6 )
     => ( ( algebr932160517623751201me_int @ P6 @ Q6 )
       => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q6 ) )
          = ( product_Pair_int_int @ P6 @ Q6 ) ) ) ) ).

% normalize_stable
thf(fact_5665_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P6: int,Q6: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( algebr932160517623751201me_int @ P6 @ Q6 ) ) ).

% normalize_coprime
thf(fact_5666_quotient__of__coprime,axiom,
    ! [R2: rat,P6: int,Q6: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( algebr932160517623751201me_int @ P6 @ Q6 ) ) ).

% quotient_of_coprime
thf(fact_5667_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_5668_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_5669_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y4: int,Z2: int] : Y4 = Z2
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y4: int,Z2: int] : Y4 = Z2
      @ pcr_rat )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_5670_quotient__of__eq,axiom,
    ! [A2: int,B2: int,P6: int,Q6: int] :
      ( ( ( quotient_of @ ( fract @ A2 @ B2 ) )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( ( fract @ P6 @ Q6 )
        = ( fract @ A2 @ B2 ) ) ) ).

% quotient_of_eq
thf(fact_5671_normalize__eq,axiom,
    ! [A2: int,B2: int,P6: int,Q6: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A2 @ B2 ) )
        = ( product_Pair_int_int @ P6 @ Q6 ) )
     => ( ( fract @ P6 @ Q6 )
        = ( fract @ A2 @ B2 ) ) ) ).

% normalize_eq
thf(fact_5672_quotient__of__Fract,axiom,
    ! [A2: int,B2: int] :
      ( ( quotient_of @ ( fract @ A2 @ B2 ) )
      = ( normalize @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).

% quotient_of_Fract
thf(fact_5673_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X2 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_5674_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F5: code_integer > option6357759511663192854e_term,D3: code_integer,I5: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F5 @ ( produc1086072967326762835nteger @ D3 @ I5 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_5675_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F5: int > option6357759511663192854e_term,D3: int,I5: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F5 @ ( product_Pair_int_int @ D3 @ I5 ) ) ) ).

% exhaustive_int'.cases
thf(fact_5676_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F5: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I5: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F5 @ ( product_Pair_int_int @ D3 @ I5 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_5677_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F5: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I5: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F5 @ ( produc1086072967326762835nteger @ D3 @ I5 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_5678_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_5679_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_5680_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% le_prod_encode_1
thf(fact_5681_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% le_prod_encode_2
thf(fact_5682_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_5683_INFM__nat,axiom,
    ! [P: nat > $o] :
      ( ( frequently_nat @ P @ cofinite_nat )
      = ( ! [M: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M @ N2 )
            & ( P @ N2 ) ) ) ) ).

% INFM_nat
thf(fact_5684_INFM__nat__inductI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ! [I5: nat] :
            ( ( P @ I5 )
           => ? [J4: nat] :
                ( ( ord_less_nat @ I5 @ J4 )
                & ( P @ J4 )
                & ( Q @ J4 ) ) )
       => ( frequently_nat @ Q @ cofinite_nat ) ) ) ).

% INFM_nat_inductI
thf(fact_5685_MOST__nat,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
      = ( ? [M: nat] :
          ! [N2: nat] :
            ( ( ord_less_nat @ M @ N2 )
           => ( P @ N2 ) ) ) ) ).

% MOST_nat
thf(fact_5686_cofinite__eq__sequentially,axiom,
    cofinite_nat = at_top_nat ).

% cofinite_eq_sequentially
thf(fact_5687_list__ex__iff__not__all__inverval__int,axiom,
    ! [P: int > $o,I: int,J: int] :
      ( ( list_ex_int @ P @ ( upto @ I @ J ) )
      = ( ~ ( all_interval_int @ ( comp_o_o_int @ (~) @ P ) @ I @ J ) ) ) ).

% list_ex_iff_not_all_inverval_int
thf(fact_5688_all__interval__int__def,axiom,
    ( all_interval_int
    = ( ^ [P2: int > $o,I4: int,J2: int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ I4 @ J2 ) )
         => ( P2 @ X2 ) ) ) ) ).

% all_interval_int_def
thf(fact_5689_list__all__iff__all__interval__int,axiom,
    ! [P: int > $o,I: int,J: int] :
      ( ( list_all_int @ P @ ( upto @ I @ J ) )
      = ( all_interval_int @ P @ I @ J ) ) ).

% list_all_iff_all_interval_int
thf(fact_5690_is__pure__assn__pure,axiom,
    ! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).

% is_pure_assn_pure
thf(fact_5691_is__pure__assn__starI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( is_pure_assn @ A2 )
     => ( ( is_pure_assn @ B2 )
       => ( is_pure_assn @ ( times_times_assn @ A2 @ B2 ) ) ) ) ).

% is_pure_assn_starI
thf(fact_5692_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_5693_is__pure__assn__basic__simps_I2_J,axiom,
    is_pure_assn @ one_one_assn ).

% is_pure_assn_basic_simps(2)
thf(fact_5694_is__pure__assn__def,axiom,
    ( is_pure_assn
    = ( ^ [A3: assn] :
        ? [P2: $o] :
          ( A3
          = ( pure_assn @ P2 ) ) ) ) ).

% is_pure_assn_def
thf(fact_5695_is__pure__assnE,axiom,
    ! [A2: assn] :
      ( ( is_pure_assn @ A2 )
     => ~ ! [P5: $o] :
            ( A2
           != ( pure_assn @ P5 ) ) ) ).

% is_pure_assnE
thf(fact_5696_is__pure__assn__basic__simps_I1_J,axiom,
    is_pure_assn @ bot_bot_assn ).

% is_pure_assn_basic_simps(1)
thf(fact_5697_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ! [M6: nat,N5: nat] :
                ( ( member_nat @ M6 @ M5 )
               => ( ( member_nat @ N5 @ M5 )
                 => ( member_nat @ ( gcd_lcm_nat @ M6 @ N5 ) @ M5 ) ) )
           => ( ( gcd_Lcm_nat @ M5 )
              = ( lattic8265883725875713057ax_nat @ M5 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat
thf(fact_5698_Lcm__nat__insert,axiom,
    ! [N: nat,M5: set_nat] :
      ( ( gcd_Lcm_nat @ ( insert_nat2 @ N @ M5 ) )
      = ( gcd_lcm_nat @ N @ ( gcd_Lcm_nat @ M5 ) ) ) ).

% Lcm_nat_insert
thf(fact_5699_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ! [M6: nat,N5: nat] :
              ( ( member_nat @ M6 @ M5 )
             => ( ( member_nat @ N5 @ M5 )
               => ( member_nat @ ( gcd_lcm_nat @ M6 @ N5 ) @ M5 ) ) )
         => ( member_nat @ ( gcd_Lcm_nat @ M5 ) @ M5 ) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
thf(fact_5700_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M2 @ N ) ) ) ) ).

% lcm_pos_nat
thf(fact_5701_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).

% Lcm_nat_set_eq_fold
thf(fact_5702_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] :
      ( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
      = ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).

% Lcm_int_set_eq_fold
thf(fact_5703_properI,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ! [As: set_nat,H: heap_e7401611519738050253t_unit] :
          ( ( P @ ( produc7507926704131184380et_nat @ H @ As ) )
         => ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) ) )
     => ( ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
            ( ( P @ ( produc7507926704131184380et_nat @ H @ As ) )
           => ( ( relH @ As @ H @ H5 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
               => ( P @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) )
       => ( proper @ P ) ) ) ).

% properI
thf(fact_5704_properD2,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat,H4: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( ( relH @ As2 @ H2 @ H4 )
         => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) )
           => ( P @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ) ) ) ).

% properD2
thf(fact_5705_bool__assn__proper_I4_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H3: produc3658429121746597890et_nat] :
              ( ( P @ H3 )
              & ( Q @ H3 ) ) ) ) ) ).

% bool_assn_proper(4)
thf(fact_5706_bool__assn__proper_I3_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H3: produc3658429121746597890et_nat] :
              ( ( P @ H3 )
              | ( Q @ H3 ) ) ) ) ) ).

% bool_assn_proper(3)
thf(fact_5707_bool__assn__proper_I2_J,axiom,
    ( proper
    @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ).

% bool_assn_proper(2)
thf(fact_5708_bool__assn__proper_I1_J,axiom,
    proper @ in_range ).

% bool_assn_proper(1)
thf(fact_5709_one__assn__proper,axiom,
    proper @ one_assn_raw ).

% one_assn_proper
thf(fact_5710_times__assn__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper @ ( times_assn_raw @ P @ Q ) ) ) ) ).

% times_assn_proper
thf(fact_5711_wand__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q ) ) ).

% wand_proper
thf(fact_5712_bool__assn__proper_I5_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( proper
        @ ^ [H3: produc3658429121746597890et_nat] :
            ( ( in_range @ H3 )
            & ~ ( P @ H3 ) ) ) ) ).

% bool_assn_proper(5)
thf(fact_5713_pure__assn__proper,axiom,
    ! [B2: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B2 ) ) ).

% pure_assn_proper
thf(fact_5714_Abs__assn__inverse,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( rep_assn @ ( abs_assn @ Y ) )
        = Y ) ) ).

% Abs_assn_inverse
thf(fact_5715_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: produc3658429121746597890et_nat > $o] :
          ( ( X
            = ( abs_assn @ Y2 ) )
         => ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).

% Abs_assn_cases
thf(fact_5716_Abs__assn__induct,axiom,
    ! [P: assn > $o,X: assn] :
      ( ! [Y2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
         => ( P @ ( abs_assn @ Y2 ) ) )
     => ( P @ X ) ) ).

% Abs_assn_induct
thf(fact_5717_Abs__assn__inject,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
       => ( ( ( abs_assn @ X )
            = ( abs_assn @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_assn_inject
thf(fact_5718_Rep__assn__induct,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
       => ( P @ Y ) ) ) ).

% Rep_assn_induct
thf(fact_5719_Rep__assn__cases,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ~ ! [X3: assn] :
            ( Y
           != ( rep_assn @ X3 ) ) ) ).

% Rep_assn_cases
thf(fact_5720_Rep__assn,axiom,
    ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).

% Rep_assn
thf(fact_5721_properD1,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% properD1
thf(fact_5722_proper__iff,axiom,
    ! [P: produc3658429121746597890et_nat > $o,As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( relH @ As2 @ H2 @ H4 )
       => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) )
         => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
            = ( P @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ) ) ) ).

% proper_iff
thf(fact_5723_proper__def,axiom,
    ( proper
    = ( ^ [P2: produc3658429121746597890et_nat > $o] :
        ! [H3: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( ( P2 @ ( produc7507926704131184380et_nat @ H3 @ As7 ) )
           => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As7 ) ) )
          & ( ( ( P2 @ ( produc7507926704131184380et_nat @ H3 @ As7 ) )
              & ( relH @ As7 @ H3 @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As7 ) ) )
           => ( P2 @ ( produc7507926704131184380et_nat @ H6 @ As7 ) ) ) ) ) ) ).

% proper_def
thf(fact_5724_list__ex__iff__not__all__inverval__nat,axiom,
    ! [P: nat > $o,I: nat,J: nat] :
      ( ( list_ex_nat @ P @ ( upt @ I @ J ) )
      = ( ~ ( all_interval_nat @ ( comp_o_o_nat @ (~) @ P ) @ I @ J ) ) ) ).

% list_ex_iff_not_all_inverval_nat
thf(fact_5725_list__all__iff__all__interval__nat,axiom,
    ! [P: nat > $o,I: nat,J: nat] :
      ( ( list_all_nat @ P @ ( upt @ I @ J ) )
      = ( all_interval_nat @ P @ I @ J ) ) ).

% list_all_iff_all_interval_nat
thf(fact_5726_all__interval__nat__def,axiom,
    ( all_interval_nat
    = ( ^ [P2: nat > $o,I4: nat,J2: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) )
         => ( P2 @ X2 ) ) ) ) ).

% all_interval_nat_def
thf(fact_5727_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nat_of_num @ ( num_of_nat @ N ) )
        = N ) ) ).

% num_of_nat_inverse
thf(fact_5728_nat__of__num__pos,axiom,
    ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).

% nat_of_num_pos
thf(fact_5729_less__num__def,axiom,
    ( ord_less_num
    = ( ^ [M: num,N2: num] : ( ord_less_nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N2 ) ) ) ) ).

% less_num_def
thf(fact_5730_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).

% natural_decr
thf(fact_5731_less__natural_Orep__eq,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_natural.rep_eq
thf(fact_5732_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V2 @ W ) )
      = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% next.simps
thf(fact_5733_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc989692990947075319atural] :
      ~ ! [F5: produc4972180933644002618e_term > option6357759511663192854e_term,D3: code_natural,I5: code_natural] :
          ( X
         != ( produc3831813291587773865atural @ F5 @ ( produc3574140220909816553atural @ D3 @ I5 ) ) ) ).

% full_exhaustive_natural'.cases
thf(fact_5734_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc8731074985263844745atural] :
      ~ ! [F5: code_natural > option6357759511663192854e_term,D3: code_natural,I5: code_natural] :
          ( X
         != ( produc2252593628808123835atural @ F5 @ ( produc3574140220909816553atural @ D3 @ I5 ) ) ) ).

% exhaustive_natural'.cases
thf(fact_5735_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B4: code_natural,I5: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ B4 @ I5 ) ) ).

% log.cases
thf(fact_5736_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S6: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V4: code_natural,W4: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V5: code_natural,W5: code_natural] : ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V4 ) @ W5 ) @ ( produc3574140220909816553atural @ V5 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W4 ) ) )
              @ ( produc6591912806276919810atural @ ( next @ S6 ) ) )
          @ S6 ) ) ) ).

% split_seed_def
thf(fact_5737_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D3: int,I5: int] :
          ( X
         != ( product_Pair_int_int @ D3 @ I5 ) ) ).

% small_lazy'.cases
thf(fact_5738_Random_Orange__def,axiom,
    ( range
    = ( ^ [K4: code_natural] :
          ( produc5538323210962509403atural
          @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
            @ ^ [L4: code_natural] :
                ( produc5538323210962509403atural @ next
                @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L4 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ one_one_Code_natural )
          @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K4 ) ) ) ) ) ).

% Random.range_def
thf(fact_5739_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_nat @ Xa @ X ) ) ).

% less_natural.abs_eq
thf(fact_5740_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( ( log @ X @ Xa )
        = Y )
     => ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) )
       => ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                  | ( ord_le5570908160329646204atural @ Xa @ X ) )
               => ( Y = one_one_Code_natural ) )
              & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                    | ( ord_le5570908160329646204atural @ Xa @ X ) )
               => ( Y
                  = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) )
           => ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) ) ) ) ) ).

% log.pelims
thf(fact_5741_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ As2 )
           => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.simps
thf(fact_5742_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ! [X3: nat] :
                ( ( member_nat @ X3 @ As )
               => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_5743_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ~ ! [X7: nat] :
                  ( ( member_nat @ X7 @ As )
                 => ( ord_less_nat @ X7 @ ( lim_Product_unit @ H ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_5744_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H @ As ) )
           => ( Y
              = ( ~ ! [X2: nat] :
                      ( ( member_nat @ X2 @ As )
                     => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_5745_new__addrs__def,axiom,
    ( hoare_new_addrs
    = ( ^ [H3: heap_e7401611519738050253t_unit,As7: set_nat,H6: heap_e7401611519738050253t_unit] :
          ( sup_sup_set_nat @ As7
          @ ( collect_nat
            @ ^ [A3: nat] :
                ( ( ord_less_eq_nat @ ( lim_Product_unit @ H3 ) @ A3 )
                & ( ord_less_nat @ A3 @ ( lim_Product_unit @ H6 ) ) ) ) ) ) ) ).

% new_addrs_def
thf(fact_5746_execute__assert_H,axiom,
    ! [P: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( P
       => ( ( heap_T875086893843062177t_unit @ ( refine_Imp_assert @ P ) @ H2 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H2 @ zero_zero_nat ) ) ) ) )
      & ( ~ P
       => ( ( heap_T875086893843062177t_unit @ ( refine_Imp_assert @ P ) @ H2 )
          = none_P9117596204409417319it_nat ) ) ) ).

% execute_assert'
thf(fact_5747_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H )
         => Phi )
     => ( hoare_8945653483474564448t_unit @ P @ ( refine_Imp_assert @ Phi )
        @ ^ [Uu2: product_unit] : P ) ) ).

% assert'_rule
thf(fact_5748_wait__def,axiom,
    ( heap_Time_wait
    = ( ^ [N2: nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H3: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) ) ) ) ).

% wait_def
thf(fact_5749_less__nat__rel,axiom,
    ( ord_less_nat
    = ( transi2163837189807498211lp_nat
      @ ^ [M: nat,N2: nat] :
          ( N2
          = ( suc @ M ) ) ) ) ).

% less_nat_rel
thf(fact_5750_type__definition__assn,axiom,
    type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).

% type_definition_assn

% Helper facts (35)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( if_filter_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( if_filter_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_T,axiom,
    ! [X: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( if_lis7173127979115136509n_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_T,axiom,
    ! [X: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( if_lis7173127979115136509n_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J_T,axiom,
    ! [X: list_P7985473006766602707_nat_o,Y: list_P7985473006766602707_nat_o] :
      ( ( if_lis2904118647880296857_nat_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J_T,axiom,
    ! [X: list_P7985473006766602707_nat_o,Y: list_P7985473006766602707_nat_o] :
      ( ( if_lis2904118647880296857_nat_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: list_P5464809261938338413at_nat,Y: list_P5464809261938338413at_nat] :
      ( ( if_lis7508161100780771379at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: list_P5464809261938338413at_nat,Y: list_P5464809261938338413at_nat] :
      ( ( if_lis7508161100780771379at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    entails @ p @ q ).

%------------------------------------------------------------------------------