%------------------------------------------------------------------------------
% File     : ITP213^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00154_004690
% 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    : 0025_Hoare_Triple_00154_004690 [Des22]

% Status   : Theorem
% Rating   : 0.90 v8.2.0, 0.92 v8.1.0
% Syntax   : Number of formulae    : 11206 (4947 unt;1462 typ;   0 def)
%            Number of atoms       : 27472 (10209 equ;   0 cnn)
%            Maximal formula atoms :   48 (   2 avg)
%            Number of connectives : 103086 (2290   ~; 424   |;1813   &;87938   @)
%                                         (   0 <=>;10621  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   6 avg)
%            Number of types       :  143 ( 142 usr)
%            Number of type conns  : 7235 (7235   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1323 (1320 usr; 103 con; 0-5 aty)
%            Number of variables   : 27832 (3429   ^;23861   !; 542   ?;27832   :)
% 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 15:20:40.159
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
    set_Pr5094982260447487303t_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_mu2057375006010111271at_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    option8963830502488799655at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_se7855581050983116737at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc7819656566062154093et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    produc2115011035271226405et_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc7822875418678951345atural: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8923325533196201883nteger: $tType ).

thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    multis2468970476368604999at_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    option4624381673175914239nt_int: $tType ).

thf(ty_n_t__Set__Oset_It__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J,type,
    set_He475150555083384525t_unit: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    list_P3521021558325789923at_int: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    list_P8198026277950538467nt_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_P5707943133018811711nt_int: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
    list_P5087981734274514673_int_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    list_P3795440434834930179_o_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_set_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc6271795597528267376eger_o: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    set_set_Code_integer: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
    heap_T5738788834812785303t_unit: $tType ).

thf(ty_n_t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    heap_e7401611519738050253t_unit: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    product_prod_num_num: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    product_prod_nat_num: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    product_prod_nat_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    product_prod_int_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Num__Onum_J_J,type,
    set_option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    option_set_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    option_set_int: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Set__Oset_It__Nat__Onat_J_J,type,
    filter_set_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Ounit_J,type,
    option_Product_unit: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    product_prod_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    product_prod_int_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    product_prod_o_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    product_prod_o_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

thf(ty_n_t__Option__Ooption_It__Assertions__Oassn_J,type,
    option_assn: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
    option_set_o: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_Itf__a_J,type,
    heap_Time_Heap_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
    set_assn: $tType ).

thf(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
    list_char: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

thf(ty_n_t__List__Olist_It__Rat__Orat_J,type,
    list_rat: $tType ).

thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

thf(ty_n_t__Code____Numeral__Onatural,type,
    code_natural: $tType ).

thf(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

thf(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Rat__Orat,type,
    rat: $tType ).

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

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

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

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (1320)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

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

thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

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

thf(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: ( produc3658429121746597890et_nat > $o ) > assn ).

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

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

thf(sy_c_Assertions_Oin__range,type,
    in_range: produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

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

thf(sy_c_Assertions_Opure__assn__raw_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a825153325127701367it_nat: $o > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Opure__assn__raw__rel_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a6022498039421069578it_nat: produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o ).

thf(sy_c_Assertions_OrelH,type,
    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_BNF__Cardinal__Arithmetic_Ocone,type,
    bNF_Cardinal_cone: set_Pr5094982260447487303t_unit ).

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__Product____Type__Ounit,type,
    bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).

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_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__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_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__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__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__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_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $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,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_If_001t__Rat__Orat,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_Int_OAbs__Integ,type,
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thf(sy_c_Int_ORep__Integ,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Ointrel,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Opcr__int,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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    sup_su6535292691877529429_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su1630790145277462993_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    sup_su4182031696650224058_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
    sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su2200014604384089602_nat_o: ( multis2468970476368604999at_nat > multis2468970476368604999at_nat > $o ) > ( multis2468970476368604999at_nat > multis2468970476368604999at_nat > $o ) > multis2468970476368604999at_nat > multis2468970476368604999at_nat > $o ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    sup_su8463660629351352368_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    sup_su6256023009775730178_nat_o: ( produc4166570645942440679at_nat > $o ) > ( produc4166570645942440679at_nat > $o ) > produc4166570645942440679at_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    sup_su362511073950362882_nat_o: ( 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 > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    sup_su798857527126471912_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    sup_su8986005896011022210_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_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_M_Eo_J,type,
    sup_su2080679670758317954_nat_o: ( produc3843707927480180839at_nat > $o ) > ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat > $o ).

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

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su7519161239522478338_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_M_Eo_J_J,type,
    sup_su2191776239474528790_nat_o: ( a > produc6653097349344004940it_nat > $o ) > ( a > produc6653097349344004940it_nat > $o ) > a > produc6653097349344004940it_nat > $o ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: int > int > int ).

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    sup_su5873096292857799009at_nat: option2498585697089621389at_nat > option2498585697089621389at_nat > option2498585697089621389at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su1419176701018045153at_nat: option2158460348403737357at_nat > option2158460348403737357at_nat > option2158460348403737357at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__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_J,type,
    sup_su2273273666271716065at_nat: option1583680563626158861at_nat > option1583680563626158861at_nat > option1583680563626158861at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
    sup_sup_Product_unit: product_unit > product_unit > product_unit ).

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

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
    sup_sup_set_num: set_num > set_num > set_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    sup_su3298353300217089135nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    sup_su8975264963432250076et_nat: set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    sup_su7128418612487073120et_nat: set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    sup_su3382966977382714213nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su3035147773818789531at_nat: set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat ).

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__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_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__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    sup_su3211442154794319707it_nat: set_Pr7098220151150636591it_nat > set_Pr7098220151150636591it_nat > set_Pr7098220151150636591it_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

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

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

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Assertions__Oassn,type,
    lattic5623431474481994958t_assn: ( assn > assn > assn ) > ( assn > assn > $o ) > ( assn > assn > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Int__Oint,type,
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thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
    lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Product____Type__Ounit,type,
    lattic5111740090977526247t_unit: ( product_unit > product_unit > product_unit ) > ( product_unit > product_unit > $o ) > ( product_unit > product_unit > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Rat__Orat,type,
    lattic5374021519246970238et_rat: ( rat > rat > rat ) > ( rat > rat > $o ) > ( rat > rat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Int__Oint_J,type,
    lattic8154731777541915528et_int: ( set_int > set_int > set_int ) > ( set_int > set_int > $o ) > ( set_int > set_int > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3109210760196336428et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).

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    lattic5174734335190411425at_nat: ( set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat ) > ( set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > $o ) > ( set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    lattic5672175589177086241at_nat: ( set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ) > ( set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ) > ( set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ) > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
    linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).

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

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

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

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

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

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    nth_Pr4561283279332054789it_nat: list_P131111800688179804it_nat > nat > produc6653097349344004940it_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    nth_Pr8687047199098054322it_nat: list_P6935614879863011209it_nat > nat > produc3260487557148687353it_nat ).

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

thf(sy_c_List_Onth_001tf__a,type,
    nth_a: list_a > nat > a ).

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

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

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

thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

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

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

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

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

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

thf(sy_c_List_Oproduct_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc6966004487978991557it_nat: list_a > list_P131111800688179804it_nat > list_P6935614879863011209it_nat ).

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

thf(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

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

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

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

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

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

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

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

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_List_Oupto,type,
    upto: int > int > list_int ).

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Misc_OEps__Opt_001t__Nat__Onat,type,
    eps_Opt_nat: ( nat > $o ) > option_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Num__Onum,type,
    eps_Opt_num: ( num > $o ) > option_num ).

thf(sy_c_Misc_OEps__Opt_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,
    eps_Op6423059015816587746et_nat: ( produc3925858234332021118et_nat > $o ) > option5190343406534369742et_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    eps_Op8301357815426737072it_nat: ( produc6653097349344004940it_nat > $o ) > option233860712434008220it_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    eps_Op2013419657081471078et_nat: ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    eps_Op2446201859369042517nt_int: ( product_prod_int_int > $o ) > option4624381673175914239nt_int ).

thf(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    eps_Op3393321821070424684it_nat: ( produc8664842809031399944it_nat > $o ) > option8956607266484857688it_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    eps_Op1741419982095519837it_nat: ( produc3260487557148687353it_nat > $o ) > option3562590408128118217it_nat ).

thf(sy_c_Misc_Obijective_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,
    biject576505603616484041nt_int: set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_Misc_Obijective_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    biject1468766312547416318et_nat: set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_Misc_Obijective_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    biject2615096655818420098et_nat: set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_Misc_Obijective_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,
    biject383251550997737151nt_int: set_Pr9222295170931077689nt_int > $o ).

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

thf(sy_c_Misc_Obijective_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    biject8592306251674886141it_nat: set_Pr7098220151150636591it_nat > $o ).

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

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

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

thf(sy_c_Misc_Odflt__None__set_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    dflt_N6592383573632408824at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ).

thf(sy_c_Misc_Omap__to__set_001t__Int__Oint_001t__Int__Oint,type,
    map_to_set_int_int: ( int > option_int ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Misc_Omap__to__set_001t__Nat__Onat_001t__Nat__Onat,type,
    map_to_set_nat_nat: ( nat > option_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Misc_Opairself_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    pairse3534876335208825186e_term: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc6576344331059438605e_term ).

thf(sy_c_Misc_Opairself__rel_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    pairse6848479906795794847e_term: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ).

thf(sy_c_Misc_Orel__of_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    rel_of5543720577181062686nt_int: ( ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int ) > ( produc7773217078559923341nt_int > $o ) > set_Pr1872883991513573699nt_int ).

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

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

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

thf(sy_c_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
    rel_of_int_int: ( int > option_int ) > ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

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

thf(sy_c_Misc_Orel__of_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    rel_of2761222537318653906it_nat: ( a > option233860712434008220it_nat ) > ( produc3260487557148687353it_nat > $o ) > set_Pr7098220151150636591it_nat ).

thf(sy_c_Misc_Oset__to__map_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,
    set_to1653371902132773855nt_int: set_Pr1872883991513573699nt_int > ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int ).

thf(sy_c_Misc_Oset__to__map_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,
    set_to2047380710992656148et_nat: set_Pr8536935166611901872et_nat > ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat ).

thf(sy_c_Misc_Oset__to__map_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,
    set_to6040779677306527128et_nat: set_Pr3286484037609594932et_nat > ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat ).

thf(sy_c_Misc_Oset__to__map_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,
    set_to6585427216324510421nt_int: set_Pr9222295170931077689nt_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int ).

thf(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Nat__Onat,type,
    set_to_map_nat_nat: set_Pr1261947904930325089at_nat > nat > option_nat ).

thf(sy_c_Misc_Oset__to__map_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    set_to2008915276733485075it_nat: set_Pr7098220151150636591it_nat > a > option233860712434008220it_nat ).

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

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

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

thf(sy_c_Misc_Othe__default_001t__Num__Onum,type,
    the_default_num: num > option_num > num ).

thf(sy_c_Misc_Othe__default_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_de2487931475039285041it_nat: produc8664842809031399944it_nat > option8956607266484857688it_nat > produc8664842809031399944it_nat ).

thf(sy_c_Misc_Othe__default_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_de542837501814275618it_nat: produc3260487557148687353it_nat > option3562590408128118217it_nat > produc3260487557148687353it_nat ).

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

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

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

thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    the_de3668382347305825784at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    uncurry_int_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    uncurry_int_int_int: ( int > int > int ) > product_prod_int_int > int ).

thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    uncurr7650761721940715016nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    uncurr7511940902602773877_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_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,
    uncurr8011562610307062878at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    add_ms2612439473150266591at_nat: product_prod_nat_nat > multis2468970476368604999at_nat > multis2468970476368604999at_nat ).

thf(sy_c_Multiset_Oms__strict,type,
    ms_strict: set_Pr8551490117392284871at_nat ).

thf(sy_c_Multiset_Oms__weak,type,
    ms_weak: set_Pr8551490117392284871at_nat ).

thf(sy_c_Multiset_Opw__leq,type,
    pw_leq: multis2468970476368604999at_nat > multis2468970476368604999at_nat > $o ).

thf(sy_c_Multiset_Oset__mset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_ms8126754132646256062at_nat: multis2468970476368604999at_nat > set_Pr1261947904930325089at_nat ).

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

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

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

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

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

thf(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
    semiri3763490453095760265atural: nat > code_natural ).

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( code_integer > code_integer ) > nat > code_integer > code_integer ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    size_s341701280123345136it_nat: list_P131111800688179804it_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Rat__Orat_J,type,
    size_size_list_rat: list_rat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s8510653225128441779at_nat: multis2468970476368604999at_nat > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s8766407808098229740it_nat: option8956607266484857688it_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s2363601347547812957it_nat: option3562590408128118217it_nat > nat ).

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: num > code_natural ).

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

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

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

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

thf(sy_c_Option_Ocombine__options_001t__Num__Onum,type,
    combine_options_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin4318129983670048329it_nat: ( produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat ) > option8956607266484857688it_nat > option8956607266484857688it_nat > option8956607266484857688it_nat ).

thf(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin5819755802923621690it_nat: ( produc3260487557148687353it_nat > produc3260487557148687353it_nat > produc3260487557148687353it_nat ) > option3562590408128118217it_nat > option3562590408128118217it_nat > option3562590408128118217it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Assertions__Oassn,type,
    none_assn: option_assn ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P3773570700014501484nt_int: option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P199884684680593241et_nat: option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P4972525538344268765et_nat: option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P1286213070022356066nt_int: option7541221861074943443nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P9117596204409417319it_nat: option8956607266484857688it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P2651198173097904984it_nat: option3562590408128118217it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Ounit,type,
    none_Product_unit: option_Product_unit ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_I_Eo_J,type,
    none_set_o: option_set_o ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
    none_set_int: option_set_int ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    none_s625347054029921090at_nat: option8963830502488799655at_nat ).

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

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

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

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P7455497367792166888nt_int: produc7773217078559923341nt_int > option4256020574406277085nt_int ).

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

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

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    some_P1054239925786823975it_nat: produc6653097349344004940it_nat > option233860712434008220it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P1914260805536162275it_nat: produc8664842809031399944it_nat > option8956607266484857688it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P7913643980934408916it_nat: produc3260487557148687353it_nat > option3562590408128118217it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Ounit,type,
    some_Product_unit: product_unit > option_Product_unit ).

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

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
    some_set_o: set_o > option_set_o ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

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

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    some_s287724117700012716at_nat: set_Pr8551490117392284871at_nat > option2498585697089621389at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    some_s147305329494351046at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    some_s579751585967276588at_nat: set_Pr8693737435421807431at_nat > option2158460348403737357at_nat ).

thf(sy_c_Option_Ooption_OSome_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,
    some_s5890477192898017836at_nat: set_Pr4329608150637261639at_nat > option1583680563626158861at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Assertions__Oassn,type,
    case_option_o_assn: $o > ( assn > $o ) > option_assn > $o ).

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

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

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

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o3267218206291580230it_nat: $o > ( produc3260487557148687353it_nat > $o ) > option3562590408128118217it_nat > $o ).

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

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    case_o223999843215110191et_int: $o > ( set_int > $o ) > option_set_int > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Onatural_001t__Num__Onum,type,
    case_o5621594795226839503al_num: code_natural > ( num > code_natural ) > option_num > code_natural ).

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

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

thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Assertions__Oassn_J_001t__Assertions__Oassn,type,
    case_o4484465799723439917n_assn: option_assn > ( assn > option_assn ) > option_assn > option_assn ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
    case_o7429725398727453821at_nat: option_nat > ( nat > option_nat ) > option_nat > option_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o6061097939050036047it_nat: option3562590408128118217it_nat > ( produc3260487557148687353it_nat > option3562590408128118217it_nat ) > option3562590408128118217it_nat > option3562590408128118217it_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Ounit_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    case_o8757249484648823045at_nat: option2498585697089621389at_nat > ( set_Pr8551490117392284871at_nat > option2498585697089621389at_nat ) > option2498585697089621389at_nat > option2498585697089621389at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    case_o311359030874850053at_nat: option8963830502488799655at_nat > ( set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ) > option8963830502488799655at_nat > option8963830502488799655at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    case_o7467001542679133189at_nat: option2158460348403737357at_nat > ( set_Pr8693737435421807431at_nat > option2158460348403737357at_nat ) > option2158460348403737357at_nat > option2158460348403737357at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__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_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,
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thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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thf(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Ounit,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Assertions__Oassn_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Ounit,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

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

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Assertions__Oassn,type,
    set_or7637083652282234053n_assn: assn > set_assn ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Code____Numeral__Ointeger,type,
    set_or5754767410780653050nteger: code_integer > set_Code_integer ).

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

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

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

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

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

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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

thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).

thf(sy_c_Wellfounded_Oaccp_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,
    accp_P8928870874622223812nt_int: ( produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ) > produc7773217078559923341nt_int > $o ).

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    accp_P8458817951426537472et_nat: ( produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o ) > produc6197004810343482825et_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    accp_P5801069581201407417et_nat: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

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

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    accp_P414730952086964626it_nat: ( produc3911288613690379145it_nat > produc3911288613690379145it_nat > $o ) > produc3911288613690379145it_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
    accp_P6263092265436569219it_nat: ( produc4453839368661128058it_nat > produc4453839368661128058it_nat > $o ) > produc4453839368661128058it_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Code____Numeral__Ointeger,type,
    finite2416775604798480986nteger: set_Pr7577011563204128103nteger ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

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

thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Omlex__prod_001t__Int__Oint,type,
    mlex_prod_int: ( int > nat ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Wellfounded_Omlex__prod_001t__Nat__Onat,type,
    mlex_prod_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).

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

thf(sy_c_member_001t__Assertions__Oassn,type,
    member_assn: assn > set_assn > $o ).

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

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

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

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

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

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

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

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member5596548051065438575it_nat: option8956607266484857688it_nat > set_op239864471688321678it_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member1310632352865511136it_nat: option3562590408128118217it_nat > set_op4617111635268443007it_nat > $o ).

thf(sy_c_member_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,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6689249552917799696at_nat: produc4166570645942440679at_nat > set_Pr8551490117392284871at_nat > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    member4307123515891402160nteger: produc6491284506569428743nteger > set_Pr7577011563204128103nteger > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: produc732395585841259969at_nat > set_Pr457366540195662369at_nat > $o ).

thf(sy_c_member_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,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member4554412811331277712it_nat: produc3260487557148687353it_nat > set_Pr7098220151150636591it_nat > $o ).

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

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

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

thf(sy_v_P,type,
    p: assn ).

thf(sy_v_Q,type,
    q: a > assn ).

thf(sy_v_R,type,
    r: assn ).

thf(sy_v_as1____,type,
    as1: set_nat ).

thf(sy_v_as2____,type,
    as2: set_nat ).

thf(sy_v_as____,type,
    as: set_nat ).

thf(sy_v_c,type,
    c: heap_Time_Heap_a ).

thf(sy_v_h_H____,type,
    h: heap_e7401611519738050253t_unit ).

thf(sy_v_h____,type,
    h2: heap_e7401611519738050253t_unit ).

thf(sy_v_r____,type,
    r2: a ).

thf(sy_v_t____,type,
    t: nat ).

% Relevant facts (9692)
thf(fact_0_DJ,axiom,
    ( ( inf_inf_set_nat @ as1 @ as2 )
    = bot_bot_set_nat ) ).

% DJ
thf(fact_1__092_060open_062_Ih_H_M_Aas2_J_A_092_060Turnstile_062_AR_092_060close_062,axiom,
    rep_assn @ r @ ( produc7507926704131184380et_nat @ h @ as2 ) ).

% \<open>(h', as2) \<Turnstile> R\<close>
thf(fact_2_M2,axiom,
    rep_assn @ r @ ( produc7507926704131184380et_nat @ h2 @ as2 ) ).

% M2
thf(fact_3_DJN,axiom,
    ( ( inf_inf_set_nat @ ( hoare_new_addrs @ h2 @ as1 @ h ) @ as2 )
    = bot_bot_set_nat ) ).

% DJN
thf(fact_4__092_060open_062new__addrs_Ah_Aas_Ah_H_A_061_Anew__addrs_Ah_Aas1_Ah_H_A_092_060union_062_Aas2_092_060close_062,axiom,
    ( ( hoare_new_addrs @ h2 @ as @ h )
    = ( sup_sup_set_nat @ ( hoare_new_addrs @ h2 @ as1 @ h ) @ as2 ) ) ).

% \<open>new_addrs h as h' = new_addrs h as1 h' \<union> as2\<close>
thf(fact_5_new__addr__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_new_addrs @ H @ As @ H )
      = As ) ).

% new_addr_refl
thf(fact_6_M1,axiom,
    rep_assn @ p @ ( produc7507926704131184380et_nat @ h2 @ as1 ) ).

% M1
thf(fact_7_MDL,axiom,
    rep_assn @ ( q @ r2 ) @ ( produc7507926704131184380et_nat @ h @ ( hoare_new_addrs @ h2 @ as1 @ h ) ) ).

% MDL
thf(fact_8__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_A_K_AR_092_060close_062,axiom,
    rep_assn @ ( times_times_assn @ p @ r ) @ ( produc7507926704131184380et_nat @ h2 @ as ) ).

% \<open>(h, as) \<Turnstile> P * R\<close>
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as1_Aas2_O_A_092_060lbrakk_062as_A_061_Aas1_A_092_060union_062_Aas2_059_Aas1_A_092_060inter_062_Aas2_A_061_A_123_125_059_A_Ih_M_Aas1_J_A_092_060Turnstile_062_AP_059_A_Ih_M_Aas2_J_A_092_060Turnstile_062_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [As1: set_nat,As2: set_nat] :
        ( ( as
          = ( sup_sup_set_nat @ As1 @ As2 ) )
       => ( ( ( inf_inf_set_nat @ As1 @ As2 )
            = bot_bot_set_nat )
         => ( ( rep_assn @ p @ ( produc7507926704131184380et_nat @ h2 @ As1 ) )
           => ~ ( rep_assn @ r @ ( produc7507926704131184380et_nat @ h2 @ As2 ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>as1 as2. \<lbrakk>as = as1 \<union> as2; as1 \<inter> as2 = {}; (h, as1) \<Turnstile> P; (h, as2) \<Turnstile> R\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_11_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_12__092_060open_062relH_Aas2_Ah_Ah_H_092_060close_062,axiom,
    relH @ as2 @ h2 @ h ).

% \<open>relH as2 h h'\<close>
thf(fact_13_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X2 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X2 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_15_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_16_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_17_prod_Oinject,axiom,
    ! [X1: a,X2: produc6653097349344004940it_nat,Y1: a,Y2: produc6653097349344004940it_nat] :
      ( ( ( produc9178034014595674355it_nat @ X1 @ X2 )
        = ( produc9178034014595674355it_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( ( produc9178034014595674355it_nat @ A @ B )
        = ( produc9178034014595674355it_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_mod__starD,axiom,
    ! [A3: assn,B3: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H )
     => ? [H1: produc3658429121746597890et_nat,H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A3 @ H1 )
          & ( rep_assn @ B3 @ H2 ) ) ) ).

% mod_starD
thf(fact_24_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_25_assms,axiom,
    hoare_hoare_triple_a @ p @ c @ q ).

% assms
thf(fact_26__092_060open_062lim_Ah_A_092_060le_062_Alim_Ah_H_092_060close_062,axiom,
    ord_less_eq_nat @ ( lim_Product_unit @ h2 ) @ ( lim_Product_unit @ h ) ).

% \<open>lim h \<le> lim h'\<close>
thf(fact_27__092_060open_062as_A_061_Aas1_A_092_060union_062_Aas2_092_060close_062,axiom,
    ( as
    = ( sup_sup_set_nat @ as1 @ as2 ) ) ).

% \<open>as = as1 \<union> as2\<close>
thf(fact_28_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_29_relH__dist__union,axiom,
    ! [As: set_nat,As3: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As @ As3 ) @ H @ H3 )
      = ( ( relH @ As @ H @ H3 )
        & ( relH @ As3 @ H @ H3 ) ) ) ).

% relH_dist_union
thf(fact_30_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H4: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P2 @ ( produc5001842942810119800et_nat @ Q2 @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ).

% times_assn_raw.cases
thf(fact_31_relH__sym,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H3 )
     => ( relH @ As @ H3 @ H ) ) ).

% relH_sym
thf(fact_32_relH__trans,axiom,
    ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H22: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H12 @ H22 )
     => ( ( relH @ As @ H22 @ H32 )
       => ( relH @ As @ H12 @ H32 ) ) ) ).

% relH_trans
thf(fact_33_mod__relH,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
      ( ( relH @ As @ H @ H3 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).

% mod_relH
thf(fact_34_hoare__triple__preI,axiom,
    ! [P: assn,C: heap_Time_Heap_a,Q: a > assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H4 )
         => ( hoare_hoare_triple_a @ P @ C @ Q ) )
     => ( hoare_hoare_triple_a @ P @ C @ Q ) ) ).

% hoare_triple_preI
thf(fact_35_mod__star__conv,axiom,
    ! [A3: assn,B3: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As12: set_nat,As22: set_nat] :
            ( ( H
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As12 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As12 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A3 @ ( produc7507926704131184380et_nat @ Hr @ As12 ) )
            & ( rep_assn @ B3 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_36_star__assnI,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,As3: set_nat] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ As3 ) )
       => ( ( ( inf_inf_set_nat @ As @ As3 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As @ As3 ) ) ) ) ) ) ).

% star_assnI
thf(fact_37_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_38_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: heap_e7401611519738050253t_unit,C2: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_39_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_40_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_41_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_42_prod__induct3,axiom,
    ! [P: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ! [A4: a,B4: heap_e7401611519738050253t_unit,C2: nat] : ( P @ ( produc9178034014595674355it_nat @ A4 @ ( produc584006145561248582it_nat @ B4 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_43_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) ) ).

% prod_cases4
thf(fact_44_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: heap_e7401611519738050253t_unit,C2: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_45_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_46_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_47_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_48_prod__cases3,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A4: a,B4: heap_e7401611519738050253t_unit,C2: nat] :
          ( Y
         != ( produc9178034014595674355it_nat @ A4 @ ( produc584006145561248582it_nat @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_49_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_50_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_51_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_52_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_53_Pair__inject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( ( produc9178034014595674355it_nat @ A @ B )
        = ( produc9178034014595674355it_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_54_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_55_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_56_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P3: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_57_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P3: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_58_prod__cases,axiom,
    ! [P: produc3260487557148687353it_nat > $o,P3: produc3260487557148687353it_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] : ( P @ ( produc9178034014595674355it_nat @ A4 @ B4 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_59_surj__pair,axiom,
    ! [P3: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P3
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_60_surj__pair,axiom,
    ! [P3: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P3
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_61_surj__pair,axiom,
    ! [P3: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P3
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_62_surj__pair,axiom,
    ! [P3: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P3
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_63_surj__pair,axiom,
    ! [P3: produc3260487557148687353it_nat] :
    ? [X3: a,Y3: produc6653097349344004940it_nat] :
      ( P3
      = ( produc9178034014595674355it_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A3: set_o] :
      ( ( collect_o
        @ ^ [X4: $o] : ( member_o @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_79_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_80_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_81_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_82_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_83_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_84_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_85_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_86_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_87_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( Y
         != ( produc9178034014595674355it_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_88_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ).

% one_assn_raw.cases
thf(fact_89_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_90_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P4: assn,Q3: assn] : ( times_times_assn @ Q3 @ P4 ) ) ) ).

% assn_times_comm
thf(fact_91_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_92_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ T @ ( inf_in4302113700860409141at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_93_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ T @ ( inf_in1697001100524423349at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_94_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ T @ ( inf_in7913087082777306421at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_95_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_96_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_97_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ S @ ( inf_in4302113700860409141at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_98_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ S @ ( inf_in1697001100524423349at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_99_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_100_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_101_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_102_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_103_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_104_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_105_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_106_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_107_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_108_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_109_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_110_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_111_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_112_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ T @ ( sup_su718114333110466843at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_113_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ T @ ( sup_su3035147773818789531at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_114_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ T @ ( sup_su5525570899277871387at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_115_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_116_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_117_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ S @ ( sup_su718114333110466843at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_118_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ S @ ( sup_su3035147773818789531at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_119_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_120_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_121_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_122_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_123_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_124_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_125_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_126_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_127_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_128_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_129_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_130_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_131_Un__empty,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ A3 @ B3 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A3 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% Un_empty
thf(fact_132_Un__empty,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ A3 @ B3 )
        = bot_bo8422036546324065075at_nat )
      = ( ( A3 = bot_bo8422036546324065075at_nat )
        & ( B3 = bot_bo8422036546324065075at_nat ) ) ) ).

% Un_empty
thf(fact_133_Un__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
        = bot_bo228742789529271731at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_134_Un__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        & ( B3 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_135_Un__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( sup_sup_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ( A3 = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_136_Un__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ( A3 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_137_Un__empty,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( sup_sup_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ( A3 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_138_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_139_inf__sup__absorb,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_140_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_141_inf__sup__absorb,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ X @ ( sup_su718114333110466843at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_142_inf__sup__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_143_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_144_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_145_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_146_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_147_sup__inf__absorb,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_148_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_149_sup__inf__absorb,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( inf_in4302113700860409141at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_150_sup__inf__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_151_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_152_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_153_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_154_sup__bot__left,axiom,
    ! [X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_155_sup__bot__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_156_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_157_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_158_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_159_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_160_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_161_empty__Collect__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( bot_bo1796632182523588997nt_int
        = ( collec213857154873943460nt_int @ P ) )
      = ( ! [X4: product_prod_int_int] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_162_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_163_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_164_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_165_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X4: $o] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_166_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_167_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_168_Collect__empty__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( ( collec213857154873943460nt_int @ P )
        = bot_bo1796632182523588997nt_int )
      = ( ! [X4: product_prod_int_int] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_169_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_170_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_171_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_172_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_173_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_174_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_175_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ! [X4: set_nat] :
            ~ ( member_set_nat @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_176_all__not__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ A3 ) )
      = ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_177_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X4: $o] :
            ~ ( member_o @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_178_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_179_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ! [X4: int] :
            ~ ( member_int @ X4 @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_180_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_181_empty__iff,axiom,
    ! [C: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_182_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_183_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_184_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_185_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_186_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_187_inf__right__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_188_inf__right__idem,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_right_idem
thf(fact_189_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_190_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_191_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ B )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.right_idem
thf(fact_192_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_193_inf_Oright__idem,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ B )
      = ( inf_inf_Product_unit @ A @ B ) ) ).

% inf.right_idem
thf(fact_194_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_195_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_196_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_197_inf__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_198_inf__left__idem,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_left_idem
thf(fact_199_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_200_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_201_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ A @ ( inf_inf_assn @ A @ B ) )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.left_idem
thf(fact_202_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_203_inf_Oleft__idem,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ A @ B ) )
      = ( inf_inf_Product_unit @ A @ B ) ) ).

% inf.left_idem
thf(fact_204_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_205_inf__idem,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_206_inf__idem,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% inf_idem
thf(fact_207_inf__idem,axiom,
    ! [X: nat] :
      ( ( inf_inf_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_208_inf__idem,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ X )
      = X ) ).

% inf_idem
thf(fact_209_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_210_inf_Oidem,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_211_inf_Oidem,axiom,
    ! [A: assn] :
      ( ( inf_inf_assn @ A @ A )
      = A ) ).

% inf.idem
thf(fact_212_inf_Oidem,axiom,
    ! [A: nat] :
      ( ( inf_inf_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_213_inf_Oidem,axiom,
    ! [A: product_unit] :
      ( ( inf_inf_Product_unit @ A @ A )
      = A ) ).

% inf.idem
thf(fact_214_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_215_sup_Oright__idem,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A @ B ) @ B )
      = ( sup_su718114333110466843at_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_216_sup_Oright__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ B )
      = ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_217_sup_Oright__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ B )
      = ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_218_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_219_sup_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
      = ( sup_sup_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_220_sup__left__idem,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ X @ Y ) )
      = ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_221_sup__left__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
      = ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_222_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_223_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_224_sup__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_225_sup_Oleft__idem,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A @ ( sup_su718114333110466843at_nat @ A @ B ) )
      = ( sup_su718114333110466843at_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_226_sup_Oleft__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ A @ B ) )
      = ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_227_sup_Oleft__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_228_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_229_sup_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
      = ( sup_sup_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_230_sup__idem,axiom,
    ! [X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_231_sup__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_232_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_233_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_234_sup__idem,axiom,
    ! [X: nat] :
      ( ( sup_sup_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_235_sup_Oidem,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_236_sup_Oidem,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_237_sup_Oidem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_238_sup_Oidem,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_239_sup_Oidem,axiom,
    ! [A: nat] :
      ( ( sup_sup_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_240_Int__iff,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B3 ) )
      = ( ( member_o @ C @ A3 )
        & ( member_o @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_241_Int__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
      = ( ( member_set_nat @ C @ A3 )
        & ( member_set_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_242_Int__iff,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
      = ( ( member_int @ C @ A3 )
        & ( member_int @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_243_Int__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( member_nat @ C @ A3 )
        & ( member_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_244_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( member8440522571783428010at_nat @ C @ A3 )
        & ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_245_IntI,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ A3 )
     => ( ( member_o @ C @ B3 )
       => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_246_IntI,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( ( member_set_nat @ C @ B3 )
       => ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_247_IntI,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ A3 )
     => ( ( member_int @ C @ B3 )
       => ( member_int @ C @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_248_IntI,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_249_IntI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A3 )
     => ( ( member8440522571783428010at_nat @ C @ B3 )
       => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_250_Un__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
      = ( ( member8440522571783428010at_nat @ C @ A3 )
        | ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_251_Un__iff,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B3 ) )
      = ( ( member_o @ C @ A3 )
        | ( member_o @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_252_Un__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
      = ( ( member_set_nat @ C @ A3 )
        | ( member_set_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_253_Un__iff,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) )
      = ( ( member_int @ C @ A3 )
        | ( member_int @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_254_Un__iff,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
      = ( ( member8206827879206165904at_nat @ C @ A3 )
        | ( member8206827879206165904at_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_255_Un__iff,axiom,
    ! [C: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
      = ( ( member6689249552917799696at_nat @ C @ A3 )
        | ( member6689249552917799696at_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_256_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
      = ( ( member8757157785044589968at_nat @ C @ A3 )
        | ( member8757157785044589968at_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_257_Un__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( ( member_nat @ C @ A3 )
        | ( member_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_258_UnCI,axiom,
    ! [C: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ C @ B3 )
       => ( member8440522571783428010at_nat @ C @ A3 ) )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_259_UnCI,axiom,
    ! [C: $o,B3: set_o,A3: set_o] :
      ( ( ~ ( member_o @ C @ B3 )
       => ( member_o @ C @ A3 ) )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_260_UnCI,axiom,
    ! [C: set_nat,B3: set_set_nat,A3: set_set_nat] :
      ( ( ~ ( member_set_nat @ C @ B3 )
       => ( member_set_nat @ C @ A3 ) )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_261_UnCI,axiom,
    ! [C: int,B3: set_int,A3: set_int] :
      ( ( ~ ( member_int @ C @ B3 )
       => ( member_int @ C @ A3 ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_262_UnCI,axiom,
    ! [C: produc859450856879609959at_nat,B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ C @ B3 )
       => ( member8206827879206165904at_nat @ C @ A3 ) )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_263_UnCI,axiom,
    ! [C: produc4166570645942440679at_nat,B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( ~ ( member6689249552917799696at_nat @ C @ B3 )
       => ( member6689249552917799696at_nat @ C @ A3 ) )
     => ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_264_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C @ B3 )
       => ( member8757157785044589968at_nat @ C @ A3 ) )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_265_UnCI,axiom,
    ! [C: nat,B3: set_nat,A3: set_nat] :
      ( ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ A3 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_266_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ H )
      = ( ( rep_assn @ P @ H )
        | ( rep_assn @ Q @ H ) ) ) ).

% mod_or_dist
thf(fact_267_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_268_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_269_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_270_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) )
      = ( ( ord_less_eq_assn @ A @ B )
        & ( ord_less_eq_assn @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_271_inf_Obounded__iff,axiom,
    ! [A: product_unit,B: product_unit,C: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
      = ( ( ord_le3221252021190050221t_unit @ A @ B )
        & ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_272_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_273_inf_Obounded__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C ) )
      = ( ( ord_less_eq_set_int @ A @ B )
        & ( ord_less_eq_set_int @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_274_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_275_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_276_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_277_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_278_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_279_le__inf__iff,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( ( ord_le3221252021190050221t_unit @ X @ Y )
        & ( ord_le3221252021190050221t_unit @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_280_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_281_le__inf__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ Y @ Z ) )
      = ( ( ord_less_eq_set_int @ X @ Y )
        & ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_282_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( ord_less_eq_rat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_283_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_284_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_285_sup_Obounded__iff,axiom,
    ! [B: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B @ C ) @ A )
      = ( ( ord_le3000389064537975527at_nat @ B @ A )
        & ( ord_le3000389064537975527at_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_286_sup_Obounded__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
      = ( ( ord_le8081472938463900775at_nat @ B @ A )
        & ( ord_le8081472938463900775at_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_287_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
      = ( ( ord_le1268244103169919719at_nat @ B @ A )
        & ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_288_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_289_sup_Obounded__iff,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A )
      = ( ( ord_less_eq_set_int @ B @ A )
        & ( ord_less_eq_set_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_290_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_291_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_292_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_293_le__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ Z )
      = ( ( ord_le3000389064537975527at_nat @ X @ Z )
        & ( ord_le3000389064537975527at_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_294_le__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
      = ( ( ord_le8081472938463900775at_nat @ X @ Z )
        & ( ord_le8081472938463900775at_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_295_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_296_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_297_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_int @ X @ Z )
        & ( ord_less_eq_set_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_298_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_rat @ X @ Z )
        & ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_299_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_300_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_301_inf__bot__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% inf_bot_right
thf(fact_302_inf__bot__right,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
      = bot_bot_Product_unit ) ).

% inf_bot_right
thf(fact_303_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_304_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_305_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_306_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_307_inf__bot__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% inf_bot_left
thf(fact_308_inf__bot__left,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
      = bot_bot_Product_unit ) ).

% inf_bot_left
thf(fact_309_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_310_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_311_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_312_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_313_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A @ bot_bo5327735625951526323at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_314_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A @ bot_bo8422036546324065075at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_315_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_316_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_317_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_318_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_319_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_320_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( sup_su718114333110466843at_nat @ A @ B ) )
      = ( ( A = bot_bo5327735625951526323at_nat )
        & ( B = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_321_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat
        = ( sup_su3035147773818789531at_nat @ A @ B ) )
      = ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_322_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_323_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ A @ B ) )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_324_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_325_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_326_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_327_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_328_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_329_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_330_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_331_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_332_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_333_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_334_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ A @ B )
        = bot_bo5327735625951526323at_nat )
      = ( ( A = bot_bo5327735625951526323at_nat )
        & ( B = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_335_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ A @ B )
        = bot_bo8422036546324065075at_nat )
      = ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_336_sup__bot_Oeq__neutr__iff,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 ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_337_sup__bot_Oeq__neutr__iff,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 ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_338_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_339_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_340_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_341_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ X @ Y )
        = bot_bo5327735625951526323at_nat )
      = ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_342_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ X @ Y )
        = bot_bo8422036546324065075at_nat )
      = ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_343_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_344_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_345_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_346_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_347_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_348_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( sup_su718114333110466843at_nat @ X @ Y ) )
      = ( ( X = bot_bo5327735625951526323at_nat )
        & ( Y = bot_bo5327735625951526323at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_349_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat
        = ( sup_su3035147773818789531at_nat @ X @ Y ) )
      = ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_350_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_351_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_352_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_353_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_354_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_355_sup__bot__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ bot_bo5327735625951526323at_nat )
      = X ) ).

% sup_bot_right
thf(fact_356_sup__bot__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ bot_bo8422036546324065075at_nat )
      = X ) ).

% sup_bot_right
thf(fact_357_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% sup_bot_right
thf(fact_358_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% sup_bot_right
thf(fact_359_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_360_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_361_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_362_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_363_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        | ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_364_bot__set__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int @ bot_bo8147686125503663512_int_o ) ) ).

% bot_set_def
thf(fact_365_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_366_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_367_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_368_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_369_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_370_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_371_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ H )
      = ( ( rep_assn @ P @ H )
        & ( rep_assn @ Q @ H ) ) ) ).

% mod_and_dist
thf(fact_372_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H ) ).

% mod_false
thf(fact_373_star__or__dist1,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A3 @ B3 ) @ C3 )
      = ( sup_sup_assn @ ( times_times_assn @ A3 @ C3 ) @ ( times_times_assn @ B3 @ C3 ) ) ) ).

% star_or_dist1
thf(fact_374_star__or__dist2,axiom,
    ! [C3: assn,A3: assn,B3: assn] :
      ( ( times_times_assn @ C3 @ ( sup_sup_assn @ A3 @ B3 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C3 @ A3 ) @ ( times_times_assn @ C3 @ B3 ) ) ) ).

% star_or_dist2
thf(fact_375_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_376_ex__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A3 ) )
      = ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_377_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X4: $o] : ( member_o @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_378_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_379_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ? [X4: int] : ( member_int @ X4 @ A3 ) )
      = ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_380_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A3 )
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_381_equals0I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y3 @ A3 )
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_382_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_383_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_384_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A3 )
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_385_equals0D,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_386_equals0D,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A3 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_387_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o @ A @ A3 ) ) ).

% equals0D
thf(fact_388_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_389_equals0D,axiom,
    ! [A3: set_int,A: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ ( member_int @ A @ A3 ) ) ).

% equals0D
thf(fact_390_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_391_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_392_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_393_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_394_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_395_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_396_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_397_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_398_inf__left__commute,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_399_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_400_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_401_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( inf_inf_assn @ B @ ( inf_inf_assn @ A @ C ) )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_402_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_403_inf_Oleft__commute,axiom,
    ! [B: product_unit,A: product_unit,C: product_unit] :
      ( ( inf_inf_Product_unit @ B @ ( inf_inf_Product_unit @ A @ C ) )
      = ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_404_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ B @ ( inf_in2572325071724192079at_nat @ A @ C ) )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_405_inf__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X4 ) ) ) ).

% inf_commute
thf(fact_406_inf__commute,axiom,
    ( inf_inf_assn
    = ( ^ [X4: assn,Y4: assn] : ( inf_inf_assn @ Y4 @ X4 ) ) ) ).

% inf_commute
thf(fact_407_inf__commute,axiom,
    ( inf_inf_nat
    = ( ^ [X4: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X4 ) ) ) ).

% inf_commute
thf(fact_408_inf__commute,axiom,
    ( inf_inf_Product_unit
    = ( ^ [X4: product_unit,Y4: product_unit] : ( inf_inf_Product_unit @ Y4 @ X4 ) ) ) ).

% inf_commute
thf(fact_409_inf__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y4 @ X4 ) ) ) ).

% inf_commute
thf(fact_410_inf_Ocommute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_411_inf_Ocommute,axiom,
    ( inf_inf_assn
    = ( ^ [A5: assn,B5: assn] : ( inf_inf_assn @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_412_inf_Ocommute,axiom,
    ( inf_inf_nat
    = ( ^ [A5: nat,B5: nat] : ( inf_inf_nat @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_413_inf_Ocommute,axiom,
    ( inf_inf_Product_unit
    = ( ^ [A5: product_unit,B5: product_unit] : ( inf_inf_Product_unit @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_414_inf_Ocommute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B5 @ A5 ) ) ) ).

% inf.commute
thf(fact_415_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_416_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_417_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_418_inf__assoc,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
      = ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_419_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_420_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_421_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ C )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ).

% inf.assoc
thf(fact_422_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_423_inf_Oassoc,axiom,
    ! [A: product_unit,B: product_unit,C: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ C )
      = ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ).

% inf.assoc
thf(fact_424_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_425_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_426_inf__sup__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X4: assn,Y4: assn] : ( inf_inf_assn @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_427_inf__sup__aci_I1_J,axiom,
    ( inf_inf_nat
    = ( ^ [X4: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_428_inf__sup__aci_I1_J,axiom,
    ( inf_inf_Product_unit
    = ( ^ [X4: product_unit,Y4: product_unit] : ( inf_inf_Product_unit @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_429_inf__sup__aci_I1_J,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_430_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_431_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_432_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_433_inf__sup__aci_I2_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
      = ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_434_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_435_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_436_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_437_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_438_inf__sup__aci_I3_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_439_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_440_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_441_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_442_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_443_inf__sup__aci_I4_J,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_444_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_445_sup__left__commute,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) )
      = ( sup_su718114333110466843at_nat @ Y @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_446_sup__left__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
      = ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_447_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_448_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_449_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_450_sup_Oleft__commute,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ B @ ( sup_su718114333110466843at_nat @ A @ C ) )
      = ( sup_su718114333110466843at_nat @ A @ ( sup_su718114333110466843at_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_451_sup_Oleft__commute,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ C ) )
      = ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_452_sup_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ C ) )
      = ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_453_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_454_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_455_sup__commute,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_456_sup__commute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_457_sup__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_458_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_459_sup__commute,axiom,
    ( sup_sup_nat
    = ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_460_sup_Ocommute,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_461_sup_Ocommute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A5: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_462_sup_Ocommute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_463_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_464_sup_Ocommute,axiom,
    ( sup_sup_nat
    = ( ^ [A5: nat,B5: nat] : ( sup_sup_nat @ B5 @ A5 ) ) ) ).

% sup.commute
thf(fact_465_sup__assoc,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ Z )
      = ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_466_sup__assoc,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
      = ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_467_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_468_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_469_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_470_sup_Oassoc,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A @ B ) @ C )
      = ( sup_su718114333110466843at_nat @ A @ ( sup_su718114333110466843at_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_471_sup_Oassoc,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ C )
      = ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_472_sup_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C )
      = ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_473_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_474_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_475_inf__sup__aci_I5_J,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_476_inf__sup__aci_I5_J,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_477_inf__sup__aci_I5_J,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_478_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_479_inf__sup__aci_I5_J,axiom,
    ( sup_sup_nat
    = ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_480_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ Z )
      = ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_481_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
      = ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_482_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_483_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_484_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_485_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) )
      = ( sup_su718114333110466843at_nat @ Y @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_486_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
      = ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_487_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_488_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_489_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_490_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( sup_su718114333110466843at_nat @ X @ Y ) )
      = ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_491_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
      = ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_492_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_493_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_494_inf__sup__aci_I8_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_495_Int__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C3 ) ) ) ).

% Int_left_commute
thf(fact_496_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ B3 @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) ) ) ).

% Int_left_commute
thf(fact_497_Int__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_498_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_499_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_500_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_501_Int__absorb,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_502_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_503_Int__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
      = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).

% Int_assoc
thf(fact_504_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ).

% Int_assoc
thf(fact_505_IntD2,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B3 ) )
     => ( member_o @ C @ B3 ) ) ).

% IntD2
thf(fact_506_IntD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
     => ( member_set_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_507_IntD2,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ( member_int @ C @ B3 ) ) ).

% IntD2
thf(fact_508_IntD2,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_509_IntD2,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_510_IntD1,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B3 ) )
     => ( member_o @ C @ A3 ) ) ).

% IntD1
thf(fact_511_IntD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
     => ( member_set_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_512_IntD1,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ( member_int @ C @ A3 ) ) ).

% IntD1
thf(fact_513_IntD1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_514_IntD1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_515_IntE,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B3 ) )
     => ~ ( ( member_o @ C @ A3 )
         => ~ ( member_o @ C @ B3 ) ) ) ).

% IntE
thf(fact_516_IntE,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_set_nat @ C @ A3 )
         => ~ ( member_set_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_517_IntE,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ~ ( ( member_int @ C @ A3 )
         => ~ ( member_int @ C @ B3 ) ) ) ).

% IntE
thf(fact_518_IntE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ~ ( member_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_519_IntE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
         => ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_520_Un__left__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) )
      = ( sup_su718114333110466843at_nat @ B3 @ ( sup_su718114333110466843at_nat @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_521_Un__left__commute,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) )
      = ( sup_su3035147773818789531at_nat @ B3 @ ( sup_su3035147773818789531at_nat @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_522_Un__left__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) )
      = ( sup_su5525570899277871387at_nat @ B3 @ ( sup_su5525570899277871387at_nat @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_523_Un__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_524_Un__left__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
      = ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_525_Un__left__absorb,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
      = ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_526_Un__left__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
      = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_527_Un__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_528_Un__commute,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_529_Un__commute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A6: set_Pr8551490117392284871at_nat,B6: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_530_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_531_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_532_Un__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_533_Un__absorb,axiom,
    ! [A3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_534_Un__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_535_Un__absorb,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_536_Un__assoc,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) ) ) ).

% Un_assoc
thf(fact_537_Un__assoc,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su3035147773818789531at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) ) ) ).

% Un_assoc
thf(fact_538_Un__assoc,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) ) ) ).

% Un_assoc
thf(fact_539_Un__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% Un_assoc
thf(fact_540_ball__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A3 )
           => ( P @ X4 ) )
        & ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ B3 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_541_ball__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,P: produc4166570645942440679at_nat > $o] :
      ( ( ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ A3 )
           => ( P @ X4 ) )
        & ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ B3 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_542_ball__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A3 )
           => ( P @ X4 ) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B3 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_543_ball__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A3 @ B3 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( P @ X4 ) )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_544_bex__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A3 )
            & ( P @ X4 ) )
        | ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ B3 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_545_bex__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,P: produc4166570645942440679at_nat > $o] :
      ( ( ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ A3 )
            & ( P @ X4 ) )
        | ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ B3 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_546_bex__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A3 )
            & ( P @ X4 ) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B3 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_547_bex__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A3 @ B3 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( P @ X4 ) )
        | ? [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_548_UnI2,axiom,
    ! [C: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ B3 )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_549_UnI2,axiom,
    ! [C: $o,B3: set_o,A3: set_o] :
      ( ( member_o @ C @ B3 )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_550_UnI2,axiom,
    ! [C: set_nat,B3: set_set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ B3 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_551_UnI2,axiom,
    ! [C: int,B3: set_int,A3: set_int] :
      ( ( member_int @ C @ B3 )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_552_UnI2,axiom,
    ! [C: produc859450856879609959at_nat,B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ B3 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_553_UnI2,axiom,
    ! [C: produc4166570645942440679at_nat,B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C @ B3 )
     => ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_554_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ B3 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_555_UnI2,axiom,
    ! [C: nat,B3: set_nat,A3: set_nat] :
      ( ( member_nat @ C @ B3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_556_UnI1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A3 )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_557_UnI1,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ A3 )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_558_UnI1,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_559_UnI1,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ A3 )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_560_UnI1,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ A3 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_561_UnI1,axiom,
    ! [C: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C @ A3 )
     => ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_562_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A3 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_563_UnI1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_564_UnE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
     => ( ~ ( member8440522571783428010at_nat @ C @ A3 )
       => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_565_UnE,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B3 ) )
     => ( ~ ( member_o @ C @ A3 )
       => ( member_o @ C @ B3 ) ) ) ).

% UnE
thf(fact_566_UnE,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
     => ( ~ ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_567_UnE,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B3 ) )
     => ( ~ ( member_int @ C @ A3 )
       => ( member_int @ C @ B3 ) ) ) ).

% UnE
thf(fact_568_UnE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
     => ( ~ ( member8206827879206165904at_nat @ C @ A3 )
       => ( member8206827879206165904at_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_569_UnE,axiom,
    ! [C: produc4166570645942440679at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
     => ( ~ ( member6689249552917799696at_nat @ C @ A3 )
       => ( member6689249552917799696at_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_570_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
     => ( ~ ( member8757157785044589968at_nat @ C @ A3 )
       => ( member8757157785044589968at_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_571_UnE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ~ ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_572_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_573_inf_OcoboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_574_inf_OcoboundedI2,axiom,
    ! [B: product_unit,C: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ C )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_575_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_576_inf_OcoboundedI2,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ C )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_577_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_578_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_579_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ B @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_580_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_581_inf_OcoboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_582_inf_OcoboundedI1,axiom,
    ! [A: product_unit,C: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ C )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_583_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_584_inf_OcoboundedI1,axiom,
    ! [A: set_int,C: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_585_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_586_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_587_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_588_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B5: set_nat,A5: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_589_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B5: assn,A5: assn] :
          ( ( inf_inf_assn @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_590_inf_Oabsorb__iff2,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [B5: product_unit,A5: product_unit] :
          ( ( inf_inf_Product_unit @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_591_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B5: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_592_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( ( inf_inf_set_int @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_593_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( inf_inf_rat @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_594_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( inf_inf_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_595_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( inf_inf_int @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_596_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_597_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( inf_inf_assn @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_598_inf_Oabsorb__iff1,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [A5: product_unit,B5: product_unit] :
          ( ( inf_inf_Product_unit @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_599_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_600_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( inf_inf_set_int @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_601_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( inf_inf_rat @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_602_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( inf_inf_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_603_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( inf_inf_int @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_604_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_605_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_606_inf_Ocobounded2,axiom,
    ! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_607_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_608_inf_Ocobounded2,axiom,
    ! [A: set_int,B: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_609_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_610_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_611_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_612_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_613_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_614_inf_Ocobounded1,axiom,
    ! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_615_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_616_inf_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_617_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_618_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_619_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_620_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( A5
          = ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_621_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B5: assn] :
          ( A5
          = ( inf_inf_assn @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_622_inf_Oorder__iff,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [A5: product_unit,B5: product_unit] :
          ( A5
          = ( inf_inf_Product_unit @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_623_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( A5
          = ( inf_in2572325071724192079at_nat @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_624_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( A5
          = ( inf_inf_set_int @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_625_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] :
          ( A5
          = ( inf_inf_rat @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_626_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( A5
          = ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_627_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( A5
          = ( inf_inf_int @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_628_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_629_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_630_inf__greatest,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( ord_le3221252021190050221t_unit @ X @ Z )
       => ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_631_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_632_inf__greatest,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Z )
       => ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_633_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Z )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_634_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_635_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_636_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_637_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ A @ C )
       => ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_638_inf_OboundedI,axiom,
    ! [A: product_unit,B: product_unit,C: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( ( ord_le3221252021190050221t_unit @ A @ C )
       => ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_639_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_640_inf_OboundedI,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ A @ C )
       => ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_641_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C )
       => ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_642_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_643_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_644_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_645_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) )
     => ~ ( ( ord_less_eq_assn @ A @ B )
         => ~ ( ord_less_eq_assn @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_646_inf_OboundedE,axiom,
    ! [A: product_unit,B: product_unit,C: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
     => ~ ( ( ord_le3221252021190050221t_unit @ A @ B )
         => ~ ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_647_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_648_inf_OboundedE,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C ) )
     => ~ ( ( ord_less_eq_set_int @ A @ B )
         => ~ ( ord_less_eq_set_int @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_649_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_650_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_651_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_652_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_653_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_654_inf__absorb2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ Y @ X )
     => ( ( inf_inf_Product_unit @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_655_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_656_inf__absorb2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( inf_inf_set_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_657_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( inf_inf_rat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_658_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_659_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_660_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_661_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_662_inf__absorb1,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( inf_inf_Product_unit @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_663_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_664_inf__absorb1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( inf_inf_set_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_665_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( inf_inf_rat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_666_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_667_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_668_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_669_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_670_inf_Oabsorb2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ A )
     => ( ( inf_inf_Product_unit @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_671_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_672_inf_Oabsorb2,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( inf_inf_set_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_673_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_674_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_675_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_676_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_677_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_678_inf_Oabsorb1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( ( inf_inf_Product_unit @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_679_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_680_inf_Oabsorb1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( inf_inf_set_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_681_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_682_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_683_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_684_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( inf_inf_set_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_685_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( inf_inf_assn @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_686_le__iff__inf,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [X4: product_unit,Y4: product_unit] :
          ( ( inf_inf_Product_unit @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_687_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_688_le__iff__inf,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( inf_inf_set_int @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_689_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( inf_inf_rat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_690_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( inf_inf_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_691_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( inf_inf_int @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_692_inf__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_set_nat @ X3 @ Z2 )
               => ( ord_less_eq_set_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_693_inf__unique,axiom,
    ! [F: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ( ord_less_eq_assn @ X3 @ Z2 )
               => ( ord_less_eq_assn @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_assn @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_694_inf__unique,axiom,
    ! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] :
              ( ( ord_le3221252021190050221t_unit @ X3 @ Y3 )
             => ( ( ord_le3221252021190050221t_unit @ X3 @ Z2 )
               => ( ord_le3221252021190050221t_unit @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_Product_unit @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_695_inf__unique,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X3 @ Y3 )
             => ( ( ord_le3146513528884898305at_nat @ X3 @ Z2 )
               => ( ord_le3146513528884898305at_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_696_inf__unique,axiom,
    ! [F: set_int > set_int > set_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( ( ord_less_eq_set_int @ X3 @ Y3 )
             => ( ( ord_less_eq_set_int @ X3 @ Z2 )
               => ( ord_less_eq_set_int @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_set_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_697_inf__unique,axiom,
    ! [F: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ( ord_less_eq_rat @ X3 @ Z2 )
               => ( ord_less_eq_rat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_rat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_698_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z2 )
               => ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_699_inf__unique,axiom,
    ! [F: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ( ord_less_eq_int @ X3 @ Z2 )
               => ( ord_less_eq_int @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_700_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( inf_inf_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_701_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( inf_inf_assn @ A @ B ) )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% inf.orderI
thf(fact_702_inf_OorderI,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( A
        = ( inf_inf_Product_unit @ A @ B ) )
     => ( ord_le3221252021190050221t_unit @ A @ B ) ) ).

% inf.orderI
thf(fact_703_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_704_inf_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A
        = ( inf_inf_set_int @ A @ B ) )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% inf.orderI
thf(fact_705_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( inf_inf_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% inf.orderI
thf(fact_706_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_707_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_708_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( A
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_709_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( A
        = ( inf_inf_assn @ A @ B ) ) ) ).

% inf.orderE
thf(fact_710_inf_OorderE,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( A
        = ( inf_inf_Product_unit @ A @ B ) ) ) ).

% inf.orderE
thf(fact_711_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_712_inf_OorderE,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( A
        = ( inf_inf_set_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_713_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( inf_inf_rat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_714_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_715_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_716_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_717_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_718_le__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ X )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_719_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_720_le__infI2,axiom,
    ! [B: set_int,X: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ X )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_721_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_722_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_723_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_eq_int @ B @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_724_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_725_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_726_le__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ X )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_727_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_728_le__infI1,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ X )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_729_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_730_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_731_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_732_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_733_inf__mono,axiom,
    ! [A: assn,C: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ ( inf_inf_assn @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_734_inf__mono,axiom,
    ! [A: product_unit,C: product_unit,B: product_unit,D2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ C )
     => ( ( ord_le3221252021190050221t_unit @ B @ D2 )
       => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ ( inf_inf_Product_unit @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_735_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ( ord_le3146513528884898305at_nat @ B @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_736_inf__mono,axiom,
    ! [A: set_int,C: set_int,B: set_int,D2: set_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ( ord_less_eq_set_int @ B @ D2 )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ ( inf_inf_set_int @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_737_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_738_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_739_inf__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_740_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X @ B )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_741_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ A )
     => ( ( ord_less_eq_assn @ X @ B )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% le_infI
thf(fact_742_le__infI,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ A )
     => ( ( ord_le3221252021190050221t_unit @ X @ B )
       => ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% le_infI
thf(fact_743_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ X @ B )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_744_le__infI,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ X @ A )
     => ( ( ord_less_eq_set_int @ X @ B )
       => ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_745_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ X @ B )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_746_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_747_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ X @ B )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_748_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A )
         => ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_749_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) )
     => ~ ( ( ord_less_eq_assn @ X @ A )
         => ~ ( ord_less_eq_assn @ X @ B ) ) ) ).

% le_infE
thf(fact_750_le__infE,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) )
     => ~ ( ( ord_le3221252021190050221t_unit @ X @ A )
         => ~ ( ord_le3221252021190050221t_unit @ X @ B ) ) ) ).

% le_infE
thf(fact_751_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X @ A )
         => ~ ( ord_le3146513528884898305at_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_752_le__infE,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ A @ B ) )
     => ~ ( ( ord_less_eq_set_int @ X @ A )
         => ~ ( ord_less_eq_set_int @ X @ B ) ) ) ).

% le_infE
thf(fact_753_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) )
     => ~ ( ( ord_less_eq_rat @ X @ A )
         => ~ ( ord_less_eq_rat @ X @ B ) ) ) ).

% le_infE
thf(fact_754_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_755_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
     => ~ ( ( ord_less_eq_int @ X @ A )
         => ~ ( ord_less_eq_int @ X @ B ) ) ) ).

% le_infE
thf(fact_756_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_757_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_758_inf__le2,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_759_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_760_inf__le2,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_761_inf__le2,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_762_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_763_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_764_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_765_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_766_inf__le1,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_767_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_768_inf__le1,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_769_inf__le1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_770_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_771_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_772_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_773_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_774_inf__sup__ord_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_775_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_776_inf__sup__ord_I1_J,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_777_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_778_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_779_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_780_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_781_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_782_inf__sup__ord_I2_J,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_783_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_784_inf__sup__ord_I2_J,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_785_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_786_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_787_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_788_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ B )
     => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_789_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C @ B )
     => ( ord_le8081472938463900775at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_790_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ B )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_791_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_792_sup_OcoboundedI2,axiom,
    ! [C: set_int,B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ C @ B )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_793_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_794_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_795_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_796_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ A )
     => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_797_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C @ A )
     => ( ord_le8081472938463900775at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_798_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_799_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_800_sup_OcoboundedI1,axiom,
    ! [C: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C @ A )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_801_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_802_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_803_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_804_sup_Oabsorb__iff2,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_805_sup_Oabsorb__iff2,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [A5: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_806_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_807_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_808_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( sup_sup_set_int @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_809_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( sup_sup_rat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_810_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( sup_sup_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_811_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( sup_sup_int @ A5 @ B5 )
          = B5 ) ) ) ).

% sup.absorb_iff2
thf(fact_812_sup_Oabsorb__iff1,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_813_sup_Oabsorb__iff1,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [B5: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_814_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_815_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B5: set_nat,A5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_816_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( ( sup_sup_set_int @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_817_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( sup_sup_rat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_818_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( sup_sup_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_819_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( sup_sup_int @ A5 @ B5 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_820_sup_Ocobounded2,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B @ ( sup_su718114333110466843at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_821_sup_Ocobounded2,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_822_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_823_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_824_sup_Ocobounded2,axiom,
    ! [B: set_int,A: set_int] : ( ord_less_eq_set_int @ B @ ( sup_sup_set_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_825_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_826_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_827_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_828_sup_Ocobounded1,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A @ ( sup_su718114333110466843at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_829_sup_Ocobounded1,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_830_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_831_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_832_sup_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : ( ord_less_eq_set_int @ A @ ( sup_sup_set_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_833_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_834_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_835_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_836_sup_Oorder__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat] :
          ( A5
          = ( sup_su718114333110466843at_nat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_837_sup_Oorder__iff,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [B5: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( A5
          = ( sup_su3035147773818789531at_nat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_838_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( A5
          = ( sup_su5525570899277871387at_nat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_839_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B5: set_nat,A5: set_nat] :
          ( A5
          = ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_840_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( A5
          = ( sup_sup_set_int @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_841_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( A5
          = ( sup_sup_rat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_842_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( A5
          = ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_843_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( A5
          = ( sup_sup_int @ A5 @ B5 ) ) ) ) ).

% sup.order_iff
thf(fact_844_sup_OboundedI,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B @ A )
     => ( ( ord_le3000389064537975527at_nat @ C @ A )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_845_sup_OboundedI,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B @ A )
     => ( ( ord_le8081472938463900775at_nat @ C @ A )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_846_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( ord_le1268244103169919719at_nat @ C @ A )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_847_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_848_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ A )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_849_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_850_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_851_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_852_sup_OboundedE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le3000389064537975527at_nat @ B @ A )
         => ~ ( ord_le3000389064537975527at_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_853_sup_OboundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le8081472938463900775at_nat @ B @ A )
         => ~ ( ord_le8081472938463900775at_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_854_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le1268244103169919719at_nat @ B @ A )
         => ~ ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_855_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_856_sup_OboundedE,axiom,
    ! [B: set_int,C: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_int @ B @ A )
         => ~ ( ord_less_eq_set_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_857_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_858_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_859_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_860_sup__absorb2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X @ Y )
     => ( ( sup_su718114333110466843at_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_861_sup__absorb2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ X @ Y )
     => ( ( sup_su3035147773818789531at_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_862_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_863_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_864_sup__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( sup_sup_set_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_865_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( sup_sup_rat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_866_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_867_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_868_sup__absorb1,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ Y @ X )
     => ( ( sup_su718114333110466843at_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_869_sup__absorb1,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ Y @ X )
     => ( ( sup_su3035147773818789531at_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_870_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_871_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_872_sup__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( sup_sup_set_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_873_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( sup_sup_rat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_874_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_875_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_876_sup_Oabsorb2,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ B )
     => ( ( sup_su718114333110466843at_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_877_sup_Oabsorb2,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A @ B )
     => ( ( sup_su3035147773818789531at_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_878_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_879_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_880_sup_Oabsorb2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( sup_sup_set_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_881_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_882_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_883_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_884_sup_Oabsorb1,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B @ A )
     => ( ( sup_su718114333110466843at_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_885_sup_Oabsorb1,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B @ A )
     => ( ( sup_su3035147773818789531at_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_886_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_887_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_888_sup_Oabsorb1,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( sup_sup_set_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_889_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_890_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_891_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_892_sup__unique,axiom,
    ! [F: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] :
              ( ( ord_le3000389064537975527at_nat @ Y3 @ X3 )
             => ( ( ord_le3000389064537975527at_nat @ Z2 @ X3 )
               => ( ord_le3000389064537975527at_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_su718114333110466843at_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_893_sup__unique,axiom,
    ! [F: set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
              ( ( ord_le8081472938463900775at_nat @ Y3 @ X3 )
             => ( ( ord_le8081472938463900775at_nat @ Z2 @ X3 )
               => ( ord_le8081472938463900775at_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_su3035147773818789531at_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_894_sup__unique,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ Y3 @ X3 )
             => ( ( ord_le1268244103169919719at_nat @ Z2 @ X3 )
               => ( ord_le1268244103169919719at_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_su5525570899277871387at_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_895_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( ord_less_eq_set_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z2 @ X3 )
               => ( ord_less_eq_set_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_896_sup__unique,axiom,
    ! [F: set_int > set_int > set_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( ( ord_less_eq_set_int @ Y3 @ X3 )
             => ( ( ord_less_eq_set_int @ Z2 @ X3 )
               => ( ord_less_eq_set_int @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_set_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_897_sup__unique,axiom,
    ! [F: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( ( ord_less_eq_rat @ Y3 @ X3 )
             => ( ( ord_less_eq_rat @ Z2 @ X3 )
               => ( ord_less_eq_rat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_rat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_898_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z2 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_899_sup__unique,axiom,
    ! [F: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( ord_less_eq_int @ Y3 @ X3 )
             => ( ( ord_less_eq_int @ Z2 @ X3 )
               => ( ord_less_eq_int @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_900_sup_OorderI,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( A
        = ( sup_su718114333110466843at_nat @ A @ B ) )
     => ( ord_le3000389064537975527at_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_901_sup_OorderI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A
        = ( sup_su3035147773818789531at_nat @ A @ B ) )
     => ( ord_le8081472938463900775at_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_902_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
     => ( ord_le1268244103169919719at_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_903_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_904_sup_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A
        = ( sup_sup_set_int @ A @ B ) )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% sup.orderI
thf(fact_905_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_906_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_907_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_908_sup_OorderE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B @ A )
     => ( A
        = ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_909_sup_OorderE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B @ A )
     => ( A
        = ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_910_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( A
        = ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_911_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_912_sup_OorderE,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( A
        = ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_913_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_914_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_915_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_916_le__iff__sup,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_917_le__iff__sup,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_918_le__iff__sup,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_919_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( sup_sup_set_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_920_le__iff__sup,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( sup_sup_set_int @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_921_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( sup_sup_rat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_922_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( sup_sup_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_923_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( sup_sup_int @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_924_sup__least,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ Y @ X )
     => ( ( ord_le3000389064537975527at_nat @ Z @ X )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_925_sup__least,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ Y @ X )
     => ( ( ord_le8081472938463900775at_nat @ Z @ X )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_926_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_927_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_928_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ Z @ X )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_929_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ Z @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_930_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_931_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_932_sup__mono,axiom,
    ! [A: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,D2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ C )
     => ( ( ord_le3000389064537975527at_nat @ B @ D2 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A @ B ) @ ( sup_su718114333110466843at_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_933_sup__mono,axiom,
    ! [A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A @ C )
     => ( ( ord_le8081472938463900775at_nat @ B @ D2 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ ( sup_su3035147773818789531at_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_934_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ C )
     => ( ( ord_le1268244103169919719at_nat @ B @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_935_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_936_sup__mono,axiom,
    ! [A: set_int,C: set_int,B: set_int,D2: set_int] :
      ( ( ord_less_eq_set_int @ A @ C )
     => ( ( ord_less_eq_set_int @ B @ D2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ ( sup_sup_set_int @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_937_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_938_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_939_sup__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_940_sup_Omono,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,D2: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ A )
     => ( ( ord_le3000389064537975527at_nat @ D2 @ B )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ C @ D2 ) @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_941_sup_Omono,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C @ A )
     => ( ( ord_le8081472938463900775at_nat @ D2 @ B )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ C @ D2 ) @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_942_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A )
     => ( ( ord_le1268244103169919719at_nat @ D2 @ B )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D2 ) @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_943_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ D2 @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_944_sup_Omono,axiom,
    ! [C: set_int,A: set_int,D2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C @ A )
     => ( ( ord_less_eq_set_int @ D2 @ B )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D2 ) @ ( sup_sup_set_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_945_sup_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D2 @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D2 ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_946_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_947_sup_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_948_le__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X @ B )
     => ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_949_le__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ X @ B )
     => ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_950_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ B )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_951_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_952_le__supI2,axiom,
    ! [X: set_int,B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ X @ B )
     => ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_953_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ X @ B )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI2
thf(fact_954_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_955_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X @ B )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_956_le__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X @ A )
     => ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_957_le__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ X @ A )
     => ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_958_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ A )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_959_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_960_le__supI1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ X @ A )
     => ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_961_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI1
thf(fact_962_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_963_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_964_sup__ge2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y @ ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_965_sup__ge2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_966_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_967_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_968_sup__ge2,axiom,
    ! [Y: set_int,X: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_969_sup__ge2,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge2
thf(fact_970_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_971_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_972_sup__ge1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_973_sup__ge1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_974_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_975_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_976_sup__ge1,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_977_sup__ge1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge1
thf(fact_978_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_979_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_980_le__supI,axiom,
    ! [A: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ X )
     => ( ( ord_le3000389064537975527at_nat @ B @ X )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_981_le__supI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A @ X )
     => ( ( ord_le8081472938463900775at_nat @ B @ X )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_982_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ X )
     => ( ( ord_le1268244103169919719at_nat @ B @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_983_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ( ord_less_eq_set_nat @ B @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_984_le__supI,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ X )
     => ( ( ord_less_eq_set_int @ B @ X )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_985_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ( ord_less_eq_rat @ B @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_986_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_987_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ( ord_less_eq_int @ B @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_988_le__supE,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A @ B ) @ X )
     => ~ ( ( ord_le3000389064537975527at_nat @ A @ X )
         => ~ ( ord_le3000389064537975527at_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_989_le__supE,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ X )
     => ~ ( ( ord_le8081472938463900775at_nat @ A @ X )
         => ~ ( ord_le8081472938463900775at_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_990_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X )
     => ~ ( ( ord_le1268244103169919719at_nat @ A @ X )
         => ~ ( ord_le1268244103169919719at_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_991_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A @ X )
         => ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_992_le__supE,axiom,
    ! [A: set_int,B: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_set_int @ A @ X )
         => ~ ( ord_less_eq_set_int @ B @ X ) ) ) ).

% le_supE
thf(fact_993_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_rat @ A @ X )
         => ~ ( ord_less_eq_rat @ B @ X ) ) ) ).

% le_supE
thf(fact_994_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_995_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_int @ A @ X )
         => ~ ( ord_less_eq_int @ B @ X ) ) ) ).

% le_supE
thf(fact_996_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_997_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_998_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_999_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_1000_inf__sup__ord_I3_J,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1001_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1002_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_1003_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_1004_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y @ ( sup_su718114333110466843at_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1005_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1006_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_1007_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_1008_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1009_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1010_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_1011_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_1012_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ! [Y4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y4 @ B3 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1013_disjoint__iff__not__equal,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A3 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ B3 )
               => ( X4 = ~ Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1014_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ B3 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1015_disjoint__iff__not__equal,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ! [Y4: int] :
                ( ( member_int @ Y4 @ B3 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1016_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_1017_Int__empty__right,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_1018_Int__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_1019_Int__empty__right,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_1020_Int__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B3 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_1021_Int__empty__left,axiom,
    ! [B3: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B3 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_1022_Int__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_1023_Int__empty__left,axiom,
    ! [B3: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B3 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_1024_disjoint__iff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ A3 @ B3 )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
           => ~ ( member_set_nat @ X4 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1025_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ~ ( member8440522571783428010at_nat @ X4 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1026_disjoint__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A3 )
           => ~ ( member_o @ X4 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1027_disjoint__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ~ ( member_nat @ X4 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1028_disjoint__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ~ ( member_int @ X4 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1029_Int__emptyI,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ~ ( member_set_nat @ X3 @ B3 ) )
     => ( ( inf_inf_set_set_nat @ A3 @ B3 )
        = bot_bot_set_set_nat ) ) ).

% Int_emptyI
thf(fact_1030_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ~ ( member8440522571783428010at_nat @ X3 @ B3 ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_1031_Int__emptyI,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ~ ( member_o @ X3 @ B3 ) )
     => ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_1032_Int__emptyI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ~ ( member_nat @ X3 @ B3 ) )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_1033_Int__emptyI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ~ ( member_int @ X3 @ B3 ) )
     => ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_1034_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_1035_sup__inf__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1036_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_1037_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ Y @ Z ) @ X )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ Y @ X ) @ ( sup_su718114333110466843at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1038_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z ) @ X )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1039_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_1040_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_1041_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
      = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1042_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_1043_sup__inf__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1044_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_1045_sup__inf__distrib1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( inf_in4302113700860409141at_nat @ Y @ Z ) )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1046_sup__inf__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1047_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_1048_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_1049_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1050_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_1051_inf__sup__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1052_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_1053_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ Y @ Z ) @ X )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ Y @ X ) @ ( inf_in4302113700860409141at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1054_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1055_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_1056_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_1057_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
      = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1058_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_1059_inf__sup__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1060_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_1061_inf__sup__distrib1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) @ ( inf_in4302113700860409141at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1062_inf__sup__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1063_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_1064_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_1065_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1066_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( sup_sup_assn @ X3 @ ( inf_inf_assn @ Y3 @ Z2 ) )
          = ( inf_inf_assn @ ( sup_sup_assn @ X3 @ Y3 ) @ ( sup_sup_assn @ X3 @ Z2 ) ) )
     => ( ( 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_1067_distrib__imp2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] :
          ( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y3 @ Z2 ) )
          = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y3 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) )
     => ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
        = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1068_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( sup_su6327502436637775413at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z2 ) )
          = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X3 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1069_distrib__imp2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ X3 @ ( inf_in4302113700860409141at_nat @ Y3 @ Z2 ) )
          = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X3 @ Y3 ) @ ( sup_su718114333110466843at_nat @ X3 @ Z2 ) ) )
     => ( ( inf_in4302113700860409141at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) )
        = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) @ ( inf_in4302113700860409141at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1070_distrib__imp2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ X3 @ ( inf_in1697001100524423349at_nat @ Y3 @ Z2 ) )
          = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X3 @ Y3 ) @ ( sup_su3035147773818789531at_nat @ X3 @ Z2 ) ) )
     => ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
        = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1071_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ Y3 @ Z2 ) )
          = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1072_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
          = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1073_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) )
          = ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) )
     => ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
        = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1074_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( inf_inf_assn @ X3 @ ( sup_sup_assn @ Y3 @ Z2 ) )
          = ( sup_sup_assn @ ( inf_inf_assn @ X3 @ Y3 ) @ ( inf_inf_assn @ X3 @ Z2 ) ) )
     => ( ( 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_1075_distrib__imp1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X3: product_unit,Y3: product_unit,Z2: product_unit] :
          ( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y3 @ Z2 ) )
          = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y3 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) )
     => ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
        = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1076_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z2 ) )
          = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1077_distrib__imp1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ! [X3: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z2: set_Pr8693737435421807431at_nat] :
          ( ( inf_in4302113700860409141at_nat @ X3 @ ( sup_su718114333110466843at_nat @ Y3 @ Z2 ) )
          = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X3 @ Y3 ) @ ( inf_in4302113700860409141at_nat @ X3 @ Z2 ) ) )
     => ( ( sup_su718114333110466843at_nat @ X @ ( inf_in4302113700860409141at_nat @ Y @ Z ) )
        = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1078_distrib__imp1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
          ( ( inf_in1697001100524423349at_nat @ X3 @ ( sup_su3035147773818789531at_nat @ Y3 @ Z2 ) )
          = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X3 @ Y3 ) @ ( inf_in1697001100524423349at_nat @ X3 @ Z2 ) ) )
     => ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
        = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1079_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( inf_in7913087082777306421at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z2 ) )
          = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1080_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
          = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) )
     => ( ( 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_1081_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y3 @ Z2 ) )
          = ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) )
     => ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
        = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1082_Un__empty__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ bot_bo5327735625951526323at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1083_Un__empty__right,axiom,
    ! [A3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ bot_bo8422036546324065075at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1084_Un__empty__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ bot_bo228742789529271731at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1085_Un__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1086_Un__empty__right,axiom,
    ! [A3: set_o] :
      ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Un_empty_right
thf(fact_1087_Un__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1088_Un__empty__right,axiom,
    ! [A3: set_int] :
      ( ( sup_sup_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Un_empty_right
thf(fact_1089_Un__empty__left,axiom,
    ! [B3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1090_Un__empty__left,axiom,
    ! [B3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1091_Un__empty__left,axiom,
    ! [B3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1092_Un__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1093_Un__empty__left,axiom,
    ! [B3: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1094_Un__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1095_Un__empty__left,axiom,
    ! [B3: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1096_Un__Int__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) @ A3 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1097_Un__Int__distrib2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B3 @ C3 ) @ A3 )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B3 @ A3 ) @ ( sup_su718114333110466843at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1098_Un__Int__distrib2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B3 @ C3 ) @ A3 )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B3 @ A3 ) @ ( sup_su3035147773818789531at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1099_Un__Int__distrib2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B3 @ C3 ) @ A3 )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ A3 ) @ ( sup_su5525570899277871387at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1100_Un__Int__distrib2,axiom,
    ! [B3: set_nat,C3: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ C3 ) @ A3 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ A3 ) @ ( sup_sup_set_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_1101_Int__Un__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) @ A3 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1102_Int__Un__distrib2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) @ A3 )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B3 @ A3 ) @ ( inf_in4302113700860409141at_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1103_Int__Un__distrib2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) @ A3 )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B3 @ A3 ) @ ( inf_in1697001100524423349at_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1104_Int__Un__distrib2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) @ A3 )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B3 @ A3 ) @ ( inf_in7913087082777306421at_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1105_Int__Un__distrib2,axiom,
    ! [B3: set_nat,C3: set_nat,A3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ C3 ) @ A3 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ A3 ) @ ( inf_inf_set_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_1106_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1107_Un__Int__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B3 @ C3 ) )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ ( sup_su718114333110466843at_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1108_Un__Int__distrib,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ ( inf_in1697001100524423349at_nat @ B3 @ C3 ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ ( sup_su3035147773818789531at_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1109_Un__Int__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B3 @ C3 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ ( sup_su5525570899277871387at_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1110_Un__Int__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_1111_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1112_Int__Un__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) @ ( inf_in4302113700860409141at_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1113_Int__Un__distrib,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) @ ( inf_in1697001100524423349at_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1114_Int__Un__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) @ ( inf_in7913087082777306421at_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1115_Int__Un__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_1116_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) ) @ ( sup_su6327502436637775413at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1117_Un__Int__crazy,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) @ ( inf_in4302113700860409141at_nat @ B3 @ C3 ) ) @ ( inf_in4302113700860409141at_nat @ C3 @ A3 ) )
      = ( inf_in4302113700860409141at_nat @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) ) @ ( sup_su718114333110466843at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1118_Un__Int__crazy,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) @ ( inf_in1697001100524423349at_nat @ B3 @ C3 ) ) @ ( inf_in1697001100524423349at_nat @ C3 @ A3 ) )
      = ( inf_in1697001100524423349at_nat @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) ) @ ( sup_su3035147773818789531at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1119_Un__Int__crazy,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) @ ( inf_in7913087082777306421at_nat @ B3 @ C3 ) ) @ ( inf_in7913087082777306421at_nat @ C3 @ A3 ) )
      = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) ) @ ( sup_su5525570899277871387at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1120_Un__Int__crazy,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) ) @ ( inf_inf_set_nat @ C3 @ A3 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ B3 @ C3 ) ) @ ( sup_sup_set_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_1121_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_1122_distrib__sup__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1123_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X @ Y ) @ ( sup_sup_rat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1124_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_1125_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_1126_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_1127_distrib__sup__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ X @ ( inf_inf_set_int @ Y @ Z ) ) @ ( inf_inf_set_int @ ( sup_sup_set_int @ X @ Y ) @ ( sup_sup_set_int @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1128_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_1129_distrib__sup__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X @ ( inf_in4302113700860409141at_nat @ Y @ Z ) ) @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1130_distrib__sup__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) ) @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1131_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_1132_distrib__inf__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) @ ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1133_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X @ Y ) @ ( inf_inf_rat @ X @ Z ) ) @ ( inf_inf_rat @ X @ ( sup_sup_rat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1134_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_1135_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_1136_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_1137_distrib__inf__le,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ ( inf_inf_set_int @ X @ Y ) @ ( inf_inf_set_int @ X @ Z ) ) @ ( inf_inf_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1138_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_1139_distrib__inf__le,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) @ ( inf_in4302113700860409141at_nat @ X @ Z ) ) @ ( inf_in4302113700860409141at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1140_distrib__inf__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) @ ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1141_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_1142_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_zero_right
thf(fact_1143_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_1144_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_1145_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_1146_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_1147_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_1148_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_zero_left
thf(fact_1149_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_1150_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_1151_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_1152_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_1153_EX,axiom,
    ( ( heap_Time_execute_a @ c @ h2 )
    = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ r2 @ ( produc584006145561248582it_nat @ h @ t ) ) ) ) ).

% EX
thf(fact_1154_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ? [As13: set_nat,As23: set_nat] :
                ( ( As4
                  = ( sup_sup_set_nat @ As13 @ As23 ) )
                & ( ( inf_inf_set_nat @ As13 @ As23 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H4 @ As13 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_1155_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ~ ? [As1: set_nat,As2: set_nat] :
                  ( ( As4
                    = ( sup_sup_set_nat @ As1 @ As2 ) )
                  & ( ( inf_inf_set_nat @ As1 @ As2 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H4 @ As1 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_1156_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 )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( Y
              = ( ~ ? [As12: set_nat,As22: set_nat] :
                      ( ( As4
                        = ( sup_sup_set_nat @ As12 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As12 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H4 @ As12 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_1157_times__assn__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( times_assn_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ? [As12: set_nat,As22: set_nat] :
            ( ( As
              = ( sup_sup_set_nat @ As12 @ As22 ) )
            & ( ( inf_inf_set_nat @ As12 @ As22 )
              = bot_bot_set_nat )
            & ( P @ ( produc7507926704131184380et_nat @ H @ As12 ) )
            & ( Q @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_1158_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_1159_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_1160_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_1161_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1162_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_1163_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_1164_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_1165_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_1166_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1167_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_1168_subset__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_1169_subset__empty,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1170_subset__empty,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1171_subset__empty,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1172_empty__subsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A3 ) ).

% empty_subsetI
thf(fact_1173_empty__subsetI,axiom,
    ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).

% empty_subsetI
thf(fact_1174_empty__subsetI,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).

% empty_subsetI
thf(fact_1175_empty__subsetI,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).

% empty_subsetI
thf(fact_1176_Int__subset__iff,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( ord_less_eq_set_nat @ C3 @ A3 )
        & ( ord_less_eq_set_nat @ C3 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1177_Int__subset__iff,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( ord_le3146513528884898305at_nat @ C3 @ A3 )
        & ( ord_le3146513528884898305at_nat @ C3 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1178_Int__subset__iff,axiom,
    ! [C3: set_int,A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ C3 @ ( inf_inf_set_int @ A3 @ B3 ) )
      = ( ( ord_less_eq_set_int @ C3 @ A3 )
        & ( ord_less_eq_set_int @ C3 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1179_Un__subset__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ C3 )
      = ( ( ord_le3000389064537975527at_nat @ A3 @ C3 )
        & ( ord_le3000389064537975527at_nat @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1180_Un__subset__iff,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ C3 )
      = ( ( ord_le8081472938463900775at_nat @ A3 @ C3 )
        & ( ord_le8081472938463900775at_nat @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1181_Un__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ C3 )
      = ( ( ord_le1268244103169919719at_nat @ A3 @ C3 )
        & ( ord_le1268244103169919719at_nat @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1182_Un__subset__iff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( ( ord_less_eq_set_nat @ A3 @ C3 )
        & ( ord_less_eq_set_nat @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1183_Un__subset__iff,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B3 ) @ C3 )
      = ( ( ord_less_eq_set_int @ A3 @ C3 )
        & ( ord_less_eq_set_int @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1184_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B5: assn] :
          ( A5
          = ( inf_inf_assn @ A5 @ B5 ) ) ) ) ).

% less_eq_assn_def
thf(fact_1185_Int__mono,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat,D3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ D3 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ C3 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_1186_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ C3 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ D3 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_1187_Int__mono,axiom,
    ! [A3: set_int,C3: set_int,B3: set_int,D3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C3 )
     => ( ( ord_less_eq_set_int @ B3 @ D3 )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ ( inf_inf_set_int @ C3 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_1188_Int__lower1,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_1189_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_1190_Int__lower1,axiom,
    ! [A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_1191_Int__lower2,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1192_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1193_Int__lower2,axiom,
    ! [A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1194_Int__absorb1,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1195_Int__absorb1,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1196_Int__absorb1,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( inf_inf_set_int @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1197_Int__absorb2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1198_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1199_Int__absorb2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( inf_inf_set_int @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1200_Int__greatest,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ A3 )
     => ( ( ord_less_eq_set_nat @ C3 @ B3 )
       => ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1201_Int__greatest,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ C3 @ B3 )
       => ( ord_le3146513528884898305at_nat @ C3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1202_Int__greatest,axiom,
    ! [C3: set_int,A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ C3 @ A3 )
     => ( ( ord_less_eq_set_int @ C3 @ B3 )
       => ( ord_less_eq_set_int @ C3 @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1203_Int__Collect__mono,axiom,
    ! [A3: set_o,B3: set_o,P: $o > $o,Q: $o > $o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B3 @ ( collect_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1204_Int__Collect__mono,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ A3 @ B3 )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A3 @ ( collec213857154873943460nt_int @ P ) ) @ ( inf_in2269163501485487111nt_int @ B3 @ ( collec213857154873943460nt_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1205_Int__Collect__mono,axiom,
    ! [A3: set_list_nat,B3: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B3 @ ( collect_list_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1206_Int__Collect__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B3 @ ( collect_set_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1207_Int__Collect__mono,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1208_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1209_Int__Collect__mono,axiom,
    ! [A3: set_int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B3 @ ( collect_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1210_Un__mono,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ C3 )
     => ( ( ord_le3000389064537975527at_nat @ B3 @ D3 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ ( sup_su718114333110466843at_nat @ C3 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_1211_Un__mono,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,D3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A3 @ C3 )
     => ( ( ord_le8081472938463900775at_nat @ B3 @ D3 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ ( sup_su3035147773818789531at_nat @ C3 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_1212_Un__mono,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ C3 )
     => ( ( ord_le1268244103169919719at_nat @ B3 @ D3 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ ( sup_su5525570899277871387at_nat @ C3 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_1213_Un__mono,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat,D3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ D3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ C3 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_1214_Un__mono,axiom,
    ! [A3: set_int,C3: set_int,B3: set_int,D3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C3 )
     => ( ( ord_less_eq_set_int @ B3 @ D3 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B3 ) @ ( sup_sup_set_int @ C3 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_1215_Un__least,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ C3 )
     => ( ( ord_le3000389064537975527at_nat @ B3 @ C3 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_1216_Un__least,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A3 @ C3 )
     => ( ( ord_le8081472938463900775at_nat @ B3 @ C3 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_1217_Un__least,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ C3 )
     => ( ( ord_le1268244103169919719at_nat @ B3 @ C3 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_1218_Un__least,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_1219_Un__least,axiom,
    ! [A3: set_int,C3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C3 )
     => ( ( ord_less_eq_set_int @ B3 @ C3 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_1220_Un__upper1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_1221_Un__upper1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_1222_Un__upper1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_1223_Un__upper1,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_1224_Un__upper1,axiom,
    ! [A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_1225_Un__upper2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B3 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_1226_Un__upper2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B3 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_1227_Un__upper2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B3 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_1228_Un__upper2,axiom,
    ! [B3: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_1229_Un__upper2,axiom,
    ! [B3: set_int,A3: set_int] : ( ord_less_eq_set_int @ B3 @ ( sup_sup_set_int @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_1230_Un__absorb1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ B3 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1231_Un__absorb1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A3 @ B3 )
     => ( ( sup_su3035147773818789531at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1232_Un__absorb1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ B3 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1233_Un__absorb1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( sup_sup_set_nat @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1234_Un__absorb1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( sup_sup_set_int @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1235_Un__absorb2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A3 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_1236_Un__absorb2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B3 @ A3 )
     => ( ( sup_su3035147773818789531at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_1237_Un__absorb2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B3 @ A3 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_1238_Un__absorb2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( sup_sup_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_1239_Un__absorb2,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( sup_sup_set_int @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_1240_subset__UnE,axiom,
    ! [C3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
     => ~ ! [A7: set_Pr8693737435421807431at_nat] :
            ( ( ord_le3000389064537975527at_nat @ A7 @ A3 )
           => ! [B7: set_Pr8693737435421807431at_nat] :
                ( ( ord_le3000389064537975527at_nat @ B7 @ B3 )
               => ( C3
                 != ( sup_su718114333110466843at_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_1241_subset__UnE,axiom,
    ! [C3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C3 @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
     => ~ ! [A7: set_Pr8551490117392284871at_nat] :
            ( ( ord_le8081472938463900775at_nat @ A7 @ A3 )
           => ! [B7: set_Pr8551490117392284871at_nat] :
                ( ( ord_le8081472938463900775at_nat @ B7 @ B3 )
               => ( C3
                 != ( sup_su3035147773818789531at_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_1242_subset__UnE,axiom,
    ! [C3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C3 @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
     => ~ ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A7 @ A3 )
           => ! [B7: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B7 @ B3 )
               => ( C3
                 != ( sup_su5525570899277871387at_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_1243_subset__UnE,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A3 )
           => ! [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ B7 @ B3 )
               => ( C3
                 != ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_1244_subset__UnE,axiom,
    ! [C3: set_int,A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ C3 @ ( sup_sup_set_int @ A3 @ B3 ) )
     => ~ ! [A7: set_int] :
            ( ( ord_less_eq_set_int @ A7 @ A3 )
           => ! [B7: set_int] :
                ( ( ord_less_eq_set_int @ B7 @ B3 )
               => ( C3
                 != ( sup_sup_set_int @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_1245_subset__Un__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1246_subset__Un__eq,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [A6: set_Pr8551490117392284871at_nat,B6: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1247_subset__Un__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1248_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1249_subset__Un__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( sup_sup_set_int @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1250_relH__subset,axiom,
    ! [Bs: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( relH @ Bs @ H @ H3 )
     => ( ( ord_less_eq_set_nat @ As @ Bs )
       => ( relH @ As @ H @ H3 ) ) ) ).

% relH_subset
thf(fact_1251_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
        = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) ) )
      = ( ord_le3146513528884898305at_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1252_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) @ C3 )
        = ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) ) )
      = ( ord_le3000389064537975527at_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1253_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) @ C3 )
        = ( inf_in1697001100524423349at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) ) )
      = ( ord_le8081472938463900775at_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1254_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) @ C3 )
        = ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) ) )
      = ( ord_le1268244103169919719at_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1255_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
        = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) )
      = ( ord_less_eq_set_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1256_Un__Int__assoc__eq,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ( sup_sup_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ C3 )
        = ( inf_inf_set_int @ A3 @ ( sup_sup_set_int @ B3 @ C3 ) ) )
      = ( ord_less_eq_set_int @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1257_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1258_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1259_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1260_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1261_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1262_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1263_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_1264_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_1265_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z3: set_int] : Y5 = Z3 )
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1266_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1267_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1268_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1269_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1270_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1271_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1272_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1273_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1274_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1275_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1276_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1277_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1278_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1279_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1280_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1281_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1282_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1283_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_1284_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_1285_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_1286_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_1287_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_1288_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_1289_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_1290_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_1291_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1292_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_1293_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_1294_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_1295_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat,B4: rat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1296_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num,B4: num] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1297_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1298_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1299_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_int,Z3: set_int] : Y5 = Z3 )
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ B5 @ A5 )
          & ( ord_less_eq_set_int @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1300_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ B5 @ A5 )
          & ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1301_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A5: num,B5: num] :
          ( ( ord_less_eq_num @ B5 @ A5 )
          & ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1302_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1303_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1304_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1305_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1306_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1307_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1308_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1309_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1310_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1311_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1312_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1313_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_1314_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1315_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1316_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1317_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1318_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1319_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z3: set_int] : Y5 = Z3 )
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1320_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
          & ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1321_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
          & ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1322_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1323_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1324_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1325_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1326_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1327_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1328_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1329_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1330_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1331_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1332_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1333_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1334_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1335_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1336_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1337_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1338_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1339_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1340_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1341_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1342_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1343_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1344_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1345_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1346_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1347_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1348_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1349_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1350_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_1351_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1352_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_1353_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1354_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1355_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1356_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1357_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1358_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1359_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1360_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1361_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1362_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1363_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1364_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1365_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1366_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1367_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1368_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1369_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1370_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1371_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1372_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1373_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1374_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1375_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_1376_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_1377_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1378_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1379_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1380_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_1381_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_1382_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3
        = ( inf_inf_set_nat @ K @ A ) )
     => ( ( inf_inf_set_nat @ A3 @ B )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1383_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: assn,K: assn,A: assn,B: assn] :
      ( ( A3
        = ( inf_inf_assn @ K @ A ) )
     => ( ( inf_inf_assn @ A3 @ B )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1384_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( inf_inf_nat @ K @ A ) )
     => ( ( inf_inf_nat @ A3 @ B )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1385_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: product_unit,K: product_unit,A: product_unit,B: product_unit] :
      ( ( A3
        = ( inf_inf_Product_unit @ K @ A ) )
     => ( ( inf_inf_Product_unit @ A3 @ B )
        = ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1386_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( inf_in2572325071724192079at_nat @ K @ A ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1387_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3
        = ( inf_inf_set_nat @ K @ B ) )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1388_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: assn,K: assn,B: assn,A: assn] :
      ( ( B3
        = ( inf_inf_assn @ K @ B ) )
     => ( ( inf_inf_assn @ A @ B3 )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1389_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( inf_inf_nat @ K @ B ) )
     => ( ( inf_inf_nat @ A @ B3 )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1390_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: product_unit,K: product_unit,B: product_unit,A: product_unit] :
      ( ( B3
        = ( inf_inf_Product_unit @ K @ B ) )
     => ( ( inf_inf_Product_unit @ A @ B3 )
        = ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1391_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( inf_in2572325071724192079at_nat @ K @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B3 )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1392_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,K: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( A3
        = ( sup_su718114333110466843at_nat @ K @ A ) )
     => ( ( sup_su718114333110466843at_nat @ A3 @ B )
        = ( sup_su718114333110466843at_nat @ K @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1393_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A3
        = ( sup_su3035147773818789531at_nat @ K @ A ) )
     => ( ( sup_su3035147773818789531at_nat @ A3 @ B )
        = ( sup_su3035147773818789531at_nat @ K @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1394_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A3
        = ( sup_su5525570899277871387at_nat @ K @ A ) )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ B )
        = ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1395_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3
        = ( sup_sup_set_nat @ K @ A ) )
     => ( ( sup_sup_set_nat @ A3 @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1396_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( sup_sup_nat @ K @ A ) )
     => ( ( sup_sup_nat @ A3 @ B )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1397_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,K: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( B3
        = ( sup_su718114333110466843at_nat @ K @ B ) )
     => ( ( sup_su718114333110466843at_nat @ A @ B3 )
        = ( sup_su718114333110466843at_nat @ K @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1398_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( B3
        = ( sup_su3035147773818789531at_nat @ K @ B ) )
     => ( ( sup_su3035147773818789531at_nat @ A @ B3 )
        = ( sup_su3035147773818789531at_nat @ K @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1399_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B3
        = ( sup_su5525570899277871387at_nat @ K @ B ) )
     => ( ( sup_su5525570899277871387at_nat @ A @ B3 )
        = ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1400_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3
        = ( sup_sup_set_nat @ K @ B ) )
     => ( ( sup_sup_set_nat @ A @ B3 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1401_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( sup_sup_nat @ K @ B ) )
     => ( ( sup_sup_nat @ A @ B3 )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1402_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_1403_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_1404_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1405_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1406_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1407_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_1408_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_1409_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_1410_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_1411_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1412_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1413_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_1414_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1415_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_1416_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1417_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ bot_bo5327735625951526323at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1418_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ bot_bo8422036546324065075at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1419_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_1420_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_1421_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_1422_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_1423_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_1424_boolean__algebra_Oconj__disj__distrib,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 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1425_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1426_boolean__algebra_Oconj__disj__distrib,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 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1427_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) @ ( inf_in4302113700860409141at_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1428_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1429_boolean__algebra_Oconj__disj__distrib,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 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1430_boolean__algebra_Oconj__disj__distrib,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 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1431_boolean__algebra_Odisj__conj__distrib,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 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1432_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1433_boolean__algebra_Odisj__conj__distrib,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 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1434_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X @ ( inf_in4302113700860409141at_nat @ Y @ Z ) )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) @ ( sup_su718114333110466843at_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1435_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1436_boolean__algebra_Odisj__conj__distrib,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 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1437_boolean__algebra_Odisj__conj__distrib,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 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1438_boolean__algebra_Oconj__disj__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 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1439_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1440_boolean__algebra_Oconj__disj__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 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1441_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ Y @ Z ) @ X )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ Y @ X ) @ ( inf_in4302113700860409141at_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1442_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1443_boolean__algebra_Oconj__disj__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 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1444_boolean__algebra_Oconj__disj__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 ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_1445_boolean__algebra_Odisj__conj__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 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1446_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1447_boolean__algebra_Odisj__conj__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 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1448_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat,X: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ Y @ Z ) @ X )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ Y @ X ) @ ( sup_su718114333110466843at_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1449_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z ) @ X )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1450_boolean__algebra_Odisj__conj__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 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1451_boolean__algebra_Odisj__conj__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 ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_1452_sup__Some,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( sup_su1419176701018045153at_nat @ ( some_s579751585967276588at_nat @ X ) @ ( some_s579751585967276588at_nat @ Y ) )
      = ( some_s579751585967276588at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_1453_sup__Some,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( sup_su5873096292857799009at_nat @ ( some_s287724117700012716at_nat @ X ) @ ( some_s287724117700012716at_nat @ Y ) )
      = ( some_s287724117700012716at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_1454_sup__Some,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su2273273666271716065at_nat @ ( some_s5890477192898017836at_nat @ X ) @ ( some_s5890477192898017836at_nat @ Y ) )
      = ( some_s5890477192898017836at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_1455_sup__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_su3598758113090618626et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( some_set_nat @ ( sup_sup_set_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_1456_sup__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( some_nat @ ( sup_sup_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_1457_inf__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_in7812914138253463912et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( some_set_nat @ ( inf_inf_set_nat @ X @ Y ) ) ) ).

% inf_Some
thf(fact_1458_inf__Some,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_option_assn @ ( some_assn @ X ) @ ( some_assn @ Y ) )
      = ( some_assn @ ( inf_inf_assn @ X @ Y ) ) ) ).

% inf_Some
thf(fact_1459_inf__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( some_nat @ ( inf_inf_nat @ X @ Y ) ) ) ).

% inf_Some
thf(fact_1460_inf__Some,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_in5800606384722953409t_unit @ ( some_Product_unit @ X ) @ ( some_Product_unit @ Y ) )
      = ( some_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) ) ) ).

% inf_Some
thf(fact_1461_inf__Some,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in777885744494410645at_nat @ ( some_s147305329494351046at_nat @ X ) @ ( some_s147305329494351046at_nat @ Y ) )
      = ( some_s147305329494351046at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) ) ) ).

% inf_Some
thf(fact_1462_less__eq__option__Some,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ X ) @ ( some_set_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1463_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1464_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1465_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1466_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1467__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_Ah_H_At_O_A_092_060lbrakk_062execute_Ac_Ah_A_061_ASome_A_Ir_M_Ah_H_M_At_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas1_Ah_H_J_A_092_060Turnstile_062_AQ_Ar_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [R2: a,H5: heap_e7401611519738050253t_unit] :
        ( ? [T2: nat] :
            ( ( heap_Time_execute_a @ c @ h2 )
            = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ T2 ) ) ) )
       => ( ( rep_assn @ ( q @ R2 ) @ ( produc7507926704131184380et_nat @ H5 @ ( hoare_new_addrs @ h2 @ as1 @ H5 ) ) )
         => ( ( relH
              @ ( collect_nat
                @ ^ [A5: nat] :
                    ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
                    & ~ ( member_nat @ A5 @ as1 ) ) )
              @ h2
              @ H5 )
           => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ h2 ) @ ( lim_Product_unit @ H5 ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>r h' t. \<lbrakk>execute c h = Some (r, h', t); (h', new_addrs h as1 h') \<Turnstile> Q r; relH {a. a < lim h \<and> a \<notin> as1} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1468_option_Oinject,axiom,
    ! [X2: produc3260487557148687353it_nat,Y2: produc3260487557148687353it_nat] :
      ( ( ( some_P7913643980934408916it_nat @ X2 )
        = ( some_P7913643980934408916it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_1469_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_1470_option_Oinject,axiom,
    ! [X2: produc8664842809031399944it_nat,Y2: produc8664842809031399944it_nat] :
      ( ( ( some_P1914260805536162275it_nat @ X2 )
        = ( some_P1914260805536162275it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_1471_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1472_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_1473_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_1474_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1475_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1476_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( ( collec213857154873943460nt_int @ P )
        = bot_bo1796632182523588997nt_int )
      = ( P = bot_bo8147686125503663512_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1477_Collect__empty__eq__bot,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( P = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1478_Collect__empty__eq__bot,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( P = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1479_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_1480_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_1481_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_1482_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_1483_pairself_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,A4: int,B4: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ A4 @ B4 ) ) ) ).

% pairself.cases
thf(fact_1484_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B @ B2 )
       => ( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
            = bot_bo2099793752762293965at_nat )
         => ( ( inf_in2572325071724192079at_nat @ A @ B )
            = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1485_disjoint__mono,axiom,
    ! [A: set_o,A2: set_o,B: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A @ A2 )
     => ( ( ord_less_eq_set_o @ B @ B2 )
       => ( ( ( inf_inf_set_o @ A2 @ B2 )
            = bot_bot_set_o )
         => ( ( inf_inf_set_o @ A @ B )
            = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
thf(fact_1486_disjoint__mono,axiom,
    ! [A: set_nat,A2: set_nat,B: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ A2 )
     => ( ( ord_less_eq_set_nat @ B @ B2 )
       => ( ( ( inf_inf_set_nat @ A2 @ B2 )
            = bot_bot_set_nat )
         => ( ( inf_inf_set_nat @ A @ B )
            = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1487_disjoint__mono,axiom,
    ! [A: set_int,A2: set_int,B: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A @ A2 )
     => ( ( ord_less_eq_set_int @ B @ B2 )
       => ( ( ( inf_inf_set_int @ A2 @ B2 )
            = bot_bot_set_int )
         => ( ( inf_inf_set_int @ A @ B )
            = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
thf(fact_1488_disjointI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ~ ( member_set_nat @ X3 @ B ) )
     => ( ( inf_inf_set_set_nat @ A @ B )
        = bot_bot_set_set_nat ) ) ).

% disjointI
thf(fact_1489_disjointI,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 ) ) ).

% disjointI
thf(fact_1490_disjointI,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 ) ) ).

% disjointI
thf(fact_1491_disjointI,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 ) ) ).

% disjointI
thf(fact_1492_disjointI,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 ) ) ).

% disjointI
thf(fact_1493_le__some__optE,axiom,
    ! [M: set_int,X: option_set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ M ) @ X )
     => ~ ! [M2: set_int] :
            ( ( X
              = ( some_set_int @ M2 ) )
           => ~ ( ord_less_eq_set_int @ M @ M2 ) ) ) ).

% le_some_optE
thf(fact_1494_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X )
     => ~ ! [M2: rat] :
            ( ( X
              = ( some_rat @ M2 ) )
           => ~ ( ord_less_eq_rat @ M @ M2 ) ) ) ).

% le_some_optE
thf(fact_1495_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X )
     => ~ ! [M2: num] :
            ( ( X
              = ( some_num @ M2 ) )
           => ~ ( ord_less_eq_num @ M @ M2 ) ) ) ).

% le_some_optE
thf(fact_1496_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X )
     => ~ ! [M2: nat] :
            ( ( X
              = ( some_nat @ M2 ) )
           => ~ ( ord_less_eq_nat @ M @ M2 ) ) ) ).

% le_some_optE
thf(fact_1497_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X )
     => ~ ! [M2: int] :
            ( ( X
              = ( some_int @ M2 ) )
           => ~ ( ord_less_eq_int @ M @ M2 ) ) ) ).

% le_some_optE
thf(fact_1498_subsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( member8440522571783428010at_nat @ X3 @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1499_subsetI,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( member_o @ X3 @ B3 ) )
     => ( ord_less_eq_set_o @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1500_subsetI,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( member_set_nat @ X3 @ B3 ) )
     => ( ord_le6893508408891458716et_nat @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1501_subsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_nat @ X3 @ B3 ) )
     => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1502_subsetI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_int @ X3 @ B3 ) )
     => ( ord_less_eq_set_int @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1503_subset__antisym,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% subset_antisym
thf(fact_1504_RH1,axiom,
    ( relH
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ as1 ) ) )
    @ h2
    @ h ) ).

% RH1
thf(fact_1505_less__option__Some,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_option_assn @ ( some_assn @ X ) @ ( some_assn @ Y ) )
      = ( ord_less_assn @ X @ Y ) ) ).

% less_option_Some
thf(fact_1506_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_1507_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_1508_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_1509_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_1510__092_060open_062relH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_092_060close_062,axiom,
    ( relH
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ as ) ) )
    @ h2
    @ h ) ).

% \<open>relH {a. a < lim h \<and> a \<notin> as} h h'\<close>
thf(fact_1511__092_060open_062as2_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    ( ord_less_eq_set_nat @ as2
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ as1 ) ) ) ) ).

% \<open>as2 \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
thf(fact_1512__092_060open_062_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
    ( ord_less_eq_set_nat
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ as ) ) )
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ as1 ) ) ) ) ).

% \<open>{a. a < lim h \<and> a \<notin> as} \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
thf(fact_1513_in__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( member8440522571783428010at_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1514_in__mono,axiom,
    ! [A3: set_o,B3: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( member_o @ X @ A3 )
       => ( member_o @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1515_in__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ( member_set_nat @ X @ A3 )
       => ( member_set_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1516_in__mono,axiom,
    ! [A3: set_nat,B3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( member_nat @ X @ A3 )
       => ( member_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1517_in__mono,axiom,
    ! [A3: set_int,B3: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( member_int @ X @ A3 )
       => ( member_int @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1518_subsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1519_subsetD,axiom,
    ! [A3: set_o,B3: set_o,C: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( member_o @ C @ A3 )
       => ( member_o @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1520_subsetD,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1521_subsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1522_subsetD,axiom,
    ! [A3: set_int,B3: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1523_equalityE,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B3 )
         => ~ ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ).

% equalityE
thf(fact_1524_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1525_subset__eq,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [X4: $o] :
          ( ( member_o @ X4 @ A6 )
         => ( member_o @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1526_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A6 )
         => ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1527_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ( member_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1528_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X4: int] :
          ( ( member_int @ X4 @ A6 )
         => ( member_int @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1529_equalityD1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => ( ord_less_eq_set_int @ A3 @ B3 ) ) ).

% equalityD1
thf(fact_1530_equalityD2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( A3 = B3 )
     => ( ord_less_eq_set_int @ B3 @ A3 ) ) ).

% equalityD2
thf(fact_1531_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [T3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T3 @ A6 )
         => ( member8440522571783428010at_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1532_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [T3: $o] :
          ( ( member_o @ T3 @ A6 )
         => ( member_o @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1533_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T3: set_nat] :
          ( ( member_set_nat @ T3 @ A6 )
         => ( member_set_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1534_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T3: nat] :
          ( ( member_nat @ T3 @ A6 )
         => ( member_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1535_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T3: int] :
          ( ( member_int @ T3 @ A6 )
         => ( member_int @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1536_subset__refl,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).

% subset_refl
thf(fact_1537_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1538_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1539_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1540_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_1541_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1542_subset__trans,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C3 )
       => ( ord_less_eq_set_int @ A3 @ C3 ) ) ) ).

% subset_trans
thf(fact_1543_set__eq__subset,axiom,
    ( ( ^ [Y5: set_int,Z3: set_int] : Y5 = Z3 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_1544_sup__Un__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su798857527126471912_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S ) )
      = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1545_sup__Un__eq,axiom,
    ! [R: set_o,S: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( sup_sup_set_o @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1546_sup__Un__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ( sup_sup_set_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ ( sup_sup_set_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1547_sup__Un__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( sup_sup_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( sup_sup_set_int @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1548_sup__Un__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su8986005896011022210_nat_o
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ R )
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ S ) )
      = ( ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ ( sup_su718114333110466843at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1549_sup__Un__eq,axiom,
    ! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su6256023009775730178_nat_o
        @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ R )
        @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ S ) )
      = ( ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1550_sup__Un__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su2080679670758317954_nat_o
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ R )
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ S ) )
      = ( ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1551_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1552_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1553_Collect__subset,axiom,
    ! [A3: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1554_Collect__subset,axiom,
    ! [A3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1555_Collect__subset,axiom,
    ! [A3: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1556_Collect__subset,axiom,
    ! [A3: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1557_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1558_Collect__subset,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ( P @ X4 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1559_inf__Int__eq,axiom,
    ! [R: set_o,S: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( inf_inf_set_o @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1560_inf__Int__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ( inf_inf_set_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ ( inf_inf_set_set_nat @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1561_inf__Int__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( inf_inf_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( inf_inf_set_int @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1562_inf__Int__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1563_inf__Int__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in5163264567034779214_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S ) )
      = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1564_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1565_sup__Un__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( sup_su1630790145277462993_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ ( sup_su7128418612487073120et_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1566_sup__Un__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( sup_su6535292691877529429_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ ( sup_su8975264963432250076et_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1567_sup__Un__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( sup_su4182031696650224058_int_o
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ ( sup_su3382966977382714213nt_int @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1568_sup__Un__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( sup_su600977994968626096_int_o
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ ( sup_su3298353300217089135nt_int @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1569_sup__Un__eq2,axiom,
    ! [R: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( sup_su2191776239474528790_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ ( sup_su3211442154794319707it_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1570_sup__Un__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su362511073950362882_nat_o
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ ( sup_su718114333110466843at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1571_sup__Un__eq2,axiom,
    ! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su2200014604384089602_nat_o
        @ ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ ( sup_su3035147773818789531at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1572_sup__Un__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ ( sup_su5525570899277871387at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1573_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1574_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X4: $o] : ( member_o @ X4 @ A6 )
          @ ^ [X4: $o] : ( member_o @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1575_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1576_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1577_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A6 )
          @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1578_inf__Int__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( inf_in3295504058751909687_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ ( inf_in1768905781608824518et_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1579_inf__Int__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( inf_in2641120393918057659_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ ( inf_in3088352823822785602et_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1580_inf__Int__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( inf_in8045883074377559328_int_o
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ ( inf_in2576786181675586251nt_int @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1581_inf__Int__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( inf_in8201999774870019094_int_o
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ ( inf_in8806012763027673301nt_int @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1582_inf__Int__eq2,axiom,
    ! [R: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( inf_in8051596381320575356_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ ( inf_in3493107508024165057it_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1583_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1584_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X4: product_prod_int_int] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1585_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X4: list_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1586_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X4: set_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1587_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1588_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X4: int] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1589_bot__empty__eq2,axiom,
    ( bot_bo5580076615179976505_nat_o
    = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ bot_bo1481135142794719944et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1590_bot__empty__eq2,axiom,
    ( bot_bo3790638025767943357_nat_o
    = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ bot_bo5635537948650799172et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1591_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ bot_bo572930865798478029nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_1592_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ bot_bo4508923176915781079nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_1593_bot__empty__eq2,axiom,
    ( bot_bo3200480807726169982_nat_o
    = ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ bot_bo8480459777671986371it_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1594_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1595_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1596_pred__equals__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1597_pred__equals__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1598_pred__equals__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1599_pred__equals__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1600_pred__equals__eq2,axiom,
    ! [R: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_1601_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1602_pred__subset__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le5966269811547037844et_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1603_pred__subset__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le4763372923235995152et_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1604_pred__subset__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ S ) )
      = ( ord_le8725513860283290265nt_int @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1605_pred__subset__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ R )
        @ ^ [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ S ) )
      = ( ord_le135402666524580259nt_int @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1606_pred__subset__eq2,axiom,
    ! [R: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( ord_le7992815620007647562_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le4478181985103859343it_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1607_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_1608_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_1609_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_1610_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_1611_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_1612_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z2: rat] :
          ( ( ord_less_rat @ X @ Z2 )
          & ( ord_less_rat @ Z2 @ Y ) ) ) ).

% dense
thf(fact_1613_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1614_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1615_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1616_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1617_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1618_order_Oasym,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order.asym
thf(fact_1619_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_1620_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_1621_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1622_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_1623_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( A = B )
     => ( ( ord_less_assn @ B @ C )
       => ( ord_less_assn @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1624_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1625_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1626_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1627_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1628_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( B = C )
       => ( ord_less_assn @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1629_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1630_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1631_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1632_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1633_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P @ Y6 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1634_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1635_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1636_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1637_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1638_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1639_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1640_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1641_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1642_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ~ ( ord_less_assn @ A @ B ) ) ).

% dual_order.asym
thf(fact_1643_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1644_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_1645_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1646_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_1647_dual__order_Oirrefl,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ A ) ).

% dual_order.irrefl
thf(fact_1648_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1649_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_1650_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1651_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_1652_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X5: nat] : ( P5 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ? [N: nat] :
          ( ( P4 @ N )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N )
             => ~ ( P4 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1653_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat] : ( P @ A4 @ A4 )
       => ( ! [A4: rat,B4: rat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1654_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: 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 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1655_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1656_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1657_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C )
       => ( ord_less_assn @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1658_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1659_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1660_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1661_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1662_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1663_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_1664_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_1665_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_1666_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_assn @ C @ B )
       => ( ord_less_assn @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1667_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1668_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1669_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1670_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1671_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1672_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1673_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1674_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1675_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1676_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1677_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1678_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1679_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1680_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1681_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1682_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1683_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1684_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1685_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_asym
thf(fact_1686_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1687_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_1688_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1689_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_1690_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1691_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_1692_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_1693_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_1694_order__less__asym_H,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order_less_asym'
thf(fact_1695_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1696_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_1697_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1698_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_1699_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_1700_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1701_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_1702_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_1703_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_1704_ord__eq__less__subst,axiom,
    ! [A: assn,F: assn > assn,B: assn,C: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1705_ord__eq__less__subst,axiom,
    ! [A: rat,F: assn > rat,B: assn,C: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1706_ord__eq__less__subst,axiom,
    ! [A: num,F: assn > num,B: assn,C: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1707_ord__eq__less__subst,axiom,
    ! [A: nat,F: assn > nat,B: assn,C: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1708_ord__eq__less__subst,axiom,
    ! [A: int,F: assn > int,B: assn,C: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1709_ord__eq__less__subst,axiom,
    ! [A: assn,F: rat > assn,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1710_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1711_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1712_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1713_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1714_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1715_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > rat,C: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1716_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > num,C: num] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1717_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > nat,C: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1718_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > int,C: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1719_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > assn,C: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1720_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1721_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1722_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1723_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1724_order__less__irrefl,axiom,
    ! [X: assn] :
      ~ ( ord_less_assn @ X @ X ) ).

% order_less_irrefl
thf(fact_1725_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_1726_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_1727_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1728_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_1729_order__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1730_order__less__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C: rat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1731_order__less__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C: num] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1732_order__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C: nat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1733_order__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C: int] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1734_order__less__subst1,axiom,
    ! [A: rat,F: assn > rat,B: assn,C: assn] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1735_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1736_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1737_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1738_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1739_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1740_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > rat,C: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1741_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > num,C: num] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1742_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1743_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1744_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1745_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1746_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1747_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1748_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1749_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_1750_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_1751_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_1752_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_1753_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_1754_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_1755_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1756_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_1757_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_1758_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_1759_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_1760_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_1761_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_1762_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_1763_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1764_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1765_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1766_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1767_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1768_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1769_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1770_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1771_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1772_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1773_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_1774_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_1775_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_1776_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_1777_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_1778_subset__Collect__conv,axiom,
    ! [S: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ S @ ( collec213857154873943460nt_int @ P ) )
      = ( ! [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ S )
           => ( P @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1779_subset__Collect__conv,axiom,
    ! [S: set_list_nat,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ S @ ( collect_list_nat @ P ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ S )
           => ( P @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1780_subset__Collect__conv,axiom,
    ! [S: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ S @ ( collect_set_nat @ P ) )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ S )
           => ( P @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1781_subset__Collect__conv,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ S )
           => ( P @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1782_subset__Collect__conv,axiom,
    ! [S: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ S @ ( collect_int @ P ) )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ S )
           => ( P @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1783_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1784_pred__subset__eq,axiom,
    ! [R: set_o,S: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ord_less_eq_set_o @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1785_pred__subset__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ord_le6893508408891458716et_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1786_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1787_pred__subset__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ord_less_eq_set_int @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1788_Set_Oempty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X4: product_prod_int_int] : $false ) ) ).

% Set.empty_def
thf(fact_1789_Set_Oempty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X4: list_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1790_Set_Oempty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X4: set_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1791_Set_Oempty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X4: product_prod_nat_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1792_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X4: $o] : $false ) ) ).

% Set.empty_def
thf(fact_1793_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% Set.empty_def
thf(fact_1794_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X4: int] : $false ) ) ).

% Set.empty_def
thf(fact_1795_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A6 )
              & ( member_o @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1796_Int__def,axiom,
    ( inf_in2269163501485487111nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X4 @ A6 )
              & ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1797_Int__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A6 )
              & ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1798_Int__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A6 )
              & ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1799_Int__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A6 )
              & ( member_int @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1800_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              & ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1801_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              & ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1802_Int__Collect,axiom,
    ! [X: $o,A3: set_o,P: $o > $o] :
      ( ( member_o @ X @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
      = ( ( member_o @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1803_Int__Collect,axiom,
    ! [X: product_prod_int_int,A3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ X @ ( inf_in2269163501485487111nt_int @ A3 @ ( collec213857154873943460nt_int @ P ) ) )
      = ( ( member5262025264175285858nt_int @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1804_Int__Collect,axiom,
    ! [X: list_nat,A3: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) )
      = ( ( member_list_nat @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1805_Int__Collect,axiom,
    ! [X: set_nat,A3: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) )
      = ( ( member_set_nat @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1806_Int__Collect,axiom,
    ! [X: int,A3: set_int,P: int > $o] :
      ( ( member_int @ X @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) )
      = ( ( member_int @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1807_Int__Collect,axiom,
    ! [X: nat,A3: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1808_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) )
      = ( ( member8440522571783428010at_nat @ X @ A3 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_1809_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A6 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1810_inf__set__def,axiom,
    ( inf_in2269163501485487111nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( inf_in3604695632404883862_int_o
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A6 )
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1811_inf__set__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( inf_inf_list_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1812_inf__set__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( inf_inf_set_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1813_inf__set__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( inf_inf_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 )
            @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1814_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1815_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1816_Collect__conj__eq,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_in2269163501485487111nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1817_Collect__conj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1818_Collect__conj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1819_Collect__conj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1820_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1821_Collect__conj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1822_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              | ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1823_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A6 )
              | ( member_o @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1824_Un__def,axiom,
    ( sup_su6024340866399070445nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X4 @ A6 )
              | ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1825_Un__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A6 )
              | ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1826_Un__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A6 )
              | ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1827_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A6 )
              | ( member_int @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1828_Un__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A6 )
              | ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1829_Un__def,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A6: set_Pr8551490117392284871at_nat,B6: set_Pr8551490117392284871at_nat] :
          ( collec5204685387357076818at_nat
          @ ^ [X4: produc4166570645942440679at_nat] :
              ( ( member6689249552917799696at_nat @ X4 @ A6 )
              | ( member6689249552917799696at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1830_Un__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ A6 )
              | ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1831_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              | ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1832_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1833_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A6 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1834_sup__set__def,axiom,
    ( sup_su6024340866399070445nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( sup_su8463660629351352368_int_o
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A6 )
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1835_sup__set__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( sup_sup_list_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1836_sup__set__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( sup_sup_set_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1837_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 )
            @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1838_sup__set__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( sup_su8986005896011022210_nat_o
            @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A6 )
            @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1839_sup__set__def,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A6: set_Pr8551490117392284871at_nat,B6: set_Pr8551490117392284871at_nat] :
          ( collec5204685387357076818at_nat
          @ ( sup_su6256023009775730178_nat_o
            @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ A6 )
            @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1840_sup__set__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ( sup_su2080679670758317954_nat_o
            @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A6 )
            @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1841_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1842_Collect__disj__eq,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su6024340866399070445nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1843_Collect__disj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1844_Collect__disj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1845_Collect__disj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1846_Collect__disj__eq,axiom,
    ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su718114333110466843at_nat @ ( collec7088162979684241874at_nat @ P ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1847_Collect__disj__eq,axiom,
    ! [P: produc4166570645942440679at_nat > $o,Q: produc4166570645942440679at_nat > $o] :
      ( ( collec5204685387357076818at_nat
        @ ^ [X4: produc4166570645942440679at_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su3035147773818789531at_nat @ ( collec5204685387357076818at_nat @ P ) @ ( collec5204685387357076818at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1848_Collect__disj__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1849_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1850_exists__leI,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ N2 )
           => ~ ( P @ N3 ) )
       => ( P @ N2 ) )
     => ? [N4: nat] :
          ( ( ord_less_eq_nat @ N4 @ N2 )
          & ( P @ N4 ) ) ) ).

% exists_leI
thf(fact_1851_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_1852_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1853_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1854_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_1855_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_1856_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_1857_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1858_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_1859_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_1860_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_1861_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1862_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1863_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > assn,C: assn] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1864_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C: assn] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1865_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C: assn] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1866_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > rat,C: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1867_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1868_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1869_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1870_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1871_order__less__le__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C: rat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1872_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1873_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1874_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1875_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1876_order__less__le__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C: num] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1877_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1878_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1879_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1880_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1881_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C: assn] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1882_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1883_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1884_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1885_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1886_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > assn,C: assn] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1887_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1888_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1889_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1890_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1891_order__le__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1892_order__le__less__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C: rat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1893_order__le__less__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C: num] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1894_order__le__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C: nat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1895_order__le__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C: int] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1896_order__le__less__subst1,axiom,
    ! [A: rat,F: assn > rat,B: assn,C: assn] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1897_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1898_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1899_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1900_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1901_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_1902_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1903_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1904_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_1905_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_1906_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_1907_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_1908_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1909_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1910_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_1911_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_1912_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_1913_order__neq__le__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( A != B )
     => ( ( ord_less_eq_assn @ A @ B )
       => ( ord_less_assn @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1914_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1915_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1916_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1917_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1918_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1919_order__le__neq__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( A != B )
       => ( ord_less_assn @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1920_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1921_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1922_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1923_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1924_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1925_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_1926_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1927_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1928_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_1929_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_1930_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_1931_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1932_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1933_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1934_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1935_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1936_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1937_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1938_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1939_order__less__le,axiom,
    ( ord_less_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( ord_less_eq_assn @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1940_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1941_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1942_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1943_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1944_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1945_order__le__less,axiom,
    ( ord_less_eq_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( ord_less_assn @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1946_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_set_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1947_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1948_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_num @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1949_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1950_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1951_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ord_less_eq_assn @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1952_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1953_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1954_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1955_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1956_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1957_order_Ostrict__implies__order,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1958_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1959_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1960_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1961_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1962_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1963_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [B5: assn,A5: assn] :
          ( ( ord_less_eq_assn @ B5 @ A5 )
          & ~ ( ord_less_eq_assn @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1964_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B5 @ A5 )
          & ~ ( ord_less_eq_set_int @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1965_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B5 @ A5 )
          & ~ ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1966_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B5: num,A5: num] :
          ( ( ord_less_eq_num @ B5 @ A5 )
          & ~ ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1967_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1968_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1969_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C @ B )
       => ( ord_less_assn @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1970_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1971_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1972_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1973_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1974_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1975_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_assn @ C @ B )
       => ( ord_less_assn @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1976_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1977_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1978_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1979_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1980_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1981_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [B5: assn,A5: assn] :
          ( ( ord_less_eq_assn @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1982_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1983_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1984_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B5: num,A5: num] :
          ( ( ord_less_eq_num @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1985_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1986_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1987_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [B5: assn,A5: assn] :
          ( ( ord_less_assn @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1988_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B5: set_int,A5: set_int] :
          ( ( ord_less_set_int @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1989_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( ord_less_rat @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1990_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B5: num,A5: num] :
          ( ( ord_less_num @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1991_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_nat @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1992_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_int @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1993_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1994_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1995_order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( ord_less_eq_assn @ A5 @ B5 )
          & ~ ( ord_less_eq_assn @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1996_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1997_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
          & ~ ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1998_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
          & ~ ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1999_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2000_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2001_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ B @ C )
       => ( ord_less_assn @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2002_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2003_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2004_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2005_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2006_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_2007_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C )
       => ( ord_less_assn @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2008_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2009_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2010_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2011_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2012_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_2013_order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( ord_less_eq_assn @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2014_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2015_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2016_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2017_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2018_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2019_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( ord_less_assn @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2020_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ( ord_less_set_int @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2021_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_less_rat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2022_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B5: num] :
          ( ( ord_less_num @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2023_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2024_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2025_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2026_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_2027_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_2028_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_2029_less__le__not__le,axiom,
    ( ord_less_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( ord_less_eq_assn @ X4 @ Y4 )
          & ~ ( ord_less_eq_assn @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2030_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ~ ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2031_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ~ ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2032_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ~ ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2033_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2034_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_2035_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y )
         => ( ord_less_eq_rat @ X3 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_2036_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z @ X3 )
         => ( ord_less_eq_rat @ Y @ X3 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_2037_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ~ ( ord_less_assn @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2038_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ~ ( ord_less_set_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2039_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2040_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2041_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2042_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2043_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2044_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2045_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2046_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2047_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2048_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2049_nless__le,axiom,
    ! [A: assn,B: assn] :
      ( ( ~ ( ord_less_assn @ A @ B ) )
      = ( ~ ( ord_less_eq_assn @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2050_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2051_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2052_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2053_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2054_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2055_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_2056_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_2057_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_2058_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_2059_leD,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ~ ( ord_less_assn @ X @ Y ) ) ).

% leD
thf(fact_2060_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_2061_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_2062_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_2063_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_2064_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_2065_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2066_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2067_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2068_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2069_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2070_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_2071_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_2072_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_2073_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_2074_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_2075_bot_Oextremum__strict,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_2076_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_2077_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2078_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_unit,C: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ C )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2079_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2080_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_assn @ B @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2081_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ B @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2082_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2083_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ B @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_2084_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2085_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_unit,C: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ C )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2086_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2087_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( ord_less_assn @ A @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2088_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2089_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2090_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_2091_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( A5
            = ( inf_inf_set_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2092_inf_Ostrict__order__iff,axiom,
    ( ord_le361264281704409273t_unit
    = ( ^ [A5: product_unit,B5: product_unit] :
          ( ( A5
            = ( inf_inf_Product_unit @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2093_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ( A5
            = ( inf_in2572325071724192079at_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2094_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( A5
            = ( inf_inf_assn @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2095_inf_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( A5
            = ( inf_inf_rat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2096_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( A5
            = ( inf_inf_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2097_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( A5
            = ( inf_inf_int @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2098_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_set_nat @ A @ B )
         => ~ ( ord_less_set_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2099_inf_Ostrict__boundedE,axiom,
    ! [A: product_unit,B: product_unit,C: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
     => ~ ( ( ord_le361264281704409273t_unit @ A @ B )
         => ~ ( ord_le361264281704409273t_unit @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2100_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A @ B )
         => ~ ( ord_le7866589430770878221at_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2101_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ ( inf_inf_assn @ B @ C ) )
     => ~ ( ( ord_less_assn @ A @ B )
         => ~ ( ord_less_assn @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2102_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_rat @ A @ B )
         => ~ ( ord_less_rat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2103_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2104_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_2105_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2106_inf_Oabsorb4,axiom,
    ! [B: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ A )
     => ( ( inf_inf_Product_unit @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2107_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2108_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2109_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2110_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2111_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_2112_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2113_inf_Oabsorb3,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ B )
     => ( ( inf_inf_Product_unit @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2114_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2115_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2116_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2117_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2118_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_2119_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2120_less__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ X )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2121_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2122_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_assn @ B @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2123_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_rat @ B @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2124_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_nat @ B @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2125_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_int @ B @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_2126_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2127_less__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ X )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2128_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2129_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_assn @ A @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2130_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_rat @ A @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2131_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_nat @ A @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2132_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_int @ A @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_2133_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ C @ B )
     => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2134_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ C @ B )
     => ( ord_le7642048601412989811at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2135_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ B )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2136_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C @ B )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2137_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ C @ B )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2138_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2139_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2140_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2141_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ C @ A )
     => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2142_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ C @ A )
     => ( ord_le7642048601412989811at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2143_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ A )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2144_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C @ A )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2145_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ C @ A )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2146_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2147_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2148_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2149_sup_Ostrict__order__iff,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat] :
          ( ( A5
            = ( sup_su718114333110466843at_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2150_sup_Ostrict__order__iff,axiom,
    ( ord_le7642048601412989811at_nat
    = ( ^ [B5: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( ( A5
            = ( sup_su3035147773818789531at_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2151_sup_Ostrict__order__iff,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( ( A5
            = ( sup_su5525570899277871387at_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2152_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B5: set_nat,A5: set_nat] :
          ( ( A5
            = ( sup_sup_set_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2153_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B5: assn,A5: assn] :
          ( ( A5
            = ( sup_sup_assn @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2154_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( A5
            = ( sup_sup_rat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2155_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( A5
            = ( sup_sup_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2156_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( A5
            = ( sup_sup_int @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2157_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ ( sup_su718114333110466843at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le6428140832669894131at_nat @ B @ A )
         => ~ ( ord_le6428140832669894131at_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2158_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le7642048601412989811at_nat @ B @ A )
         => ~ ( ord_le7642048601412989811at_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2159_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le2604355607129572851at_nat @ B @ A )
         => ~ ( ord_le2604355607129572851at_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2160_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2161_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B @ C ) @ A )
     => ~ ( ( ord_less_assn @ B @ A )
         => ~ ( ord_less_assn @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2162_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2163_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2164_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_2165_sup_Oabsorb4,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A @ B )
     => ( ( sup_su718114333110466843at_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2166_sup_Oabsorb4,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ A @ B )
     => ( ( sup_su3035147773818789531at_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2167_sup_Oabsorb4,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A @ B )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2168_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2169_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2170_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2171_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2172_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_2173_sup_Oabsorb3,axiom,
    ! [B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ B @ A )
     => ( ( sup_su718114333110466843at_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2174_sup_Oabsorb3,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ B @ A )
     => ( ( sup_su3035147773818789531at_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2175_sup_Oabsorb3,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ B @ A )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2176_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2177_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2178_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2179_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2180_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_2181_less__supI2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ X @ B )
     => ( ord_le6428140832669894131at_nat @ X @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2182_less__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ X @ B )
     => ( ord_le7642048601412989811at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2183_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ B )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2184_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X @ B )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2185_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ X @ B )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI2
thf(fact_2186_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ X @ B )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2187_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X @ B )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_2188_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_int @ X @ B )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI2
thf(fact_2189_less__supI1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ X @ A )
     => ( ord_le6428140832669894131at_nat @ X @ ( sup_su718114333110466843at_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2190_less__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ X @ A )
     => ( ord_le7642048601412989811at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2191_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ A )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2192_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ X @ A )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2193_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ X @ A )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI1
thf(fact_2194_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2195_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_2196_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_int @ X @ A )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI1
thf(fact_2197_new__addrs__def,axiom,
    ( hoare_new_addrs
    = ( ^ [H6: heap_e7401611519738050253t_unit,As5: set_nat,H7: heap_e7401611519738050253t_unit] :
          ( sup_sup_set_nat @ As5
          @ ( collect_nat
            @ ^ [A5: nat] :
                ( ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ A5 )
                & ( ord_less_nat @ A5 @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ) ).

% new_addrs_def
thf(fact_2198_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D2: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_set_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_2199_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_rat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_2200_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_num @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_2201_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_2202_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_2203_bex2I,axiom,
    ! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2204_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2205_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2206_bex2I,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2207_bex2I,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2208_bex2I,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,S: set_Pr7098220151150636591it_nat,P: a > produc6653097349344004940it_nat > $o] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ S )
     => ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A4: a,B4: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A4 @ B4 ) @ S )
            & ( P @ A4 @ B4 ) ) ) ) ).

% bex2I
thf(fact_2209_subrelI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2210_subrelI,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R3 )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le5966269811547037844et_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2211_subrelI,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R3 )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le4763372923235995152et_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2212_subrelI,axiom,
    ! [R3: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R3 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le8725513860283290265nt_int @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2213_subrelI,axiom,
    ! [R3: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R3 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le135402666524580259nt_int @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2214_subrelI,axiom,
    ! [R3: set_Pr7098220151150636591it_nat,S2: set_Pr7098220151150636591it_nat] :
      ( ! [X3: a,Y3: produc6653097349344004940it_nat] :
          ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X3 @ Y3 ) @ R3 )
         => ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le4478181985103859343it_nat @ R3 @ S2 ) ) ).

% subrelI
thf(fact_2215_memb__imp__not__empty,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( ( member_set_nat @ X @ S )
     => ( S != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_2216_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_2217_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_2218_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_2219_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_2220_set__notEmptyE,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ~ ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_2221_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_2222_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_2223_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_2224_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_2225_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ S3 )
     => ( ( ( inf_inf_set_nat @ A3 @ S3 )
          = ( inf_inf_set_nat @ B3 @ S3 ) )
       => ( ( inf_inf_set_nat @ A3 @ S )
          = ( inf_inf_set_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2226_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ S3 )
     => ( ( ( inf_in2572325071724192079at_nat @ A3 @ S3 )
          = ( inf_in2572325071724192079at_nat @ B3 @ S3 ) )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ S )
          = ( inf_in2572325071724192079at_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2227_inter__eq__subsetI,axiom,
    ! [S: set_int,S3: set_int,A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ S @ S3 )
     => ( ( ( inf_inf_set_int @ A3 @ S3 )
          = ( inf_inf_set_int @ B3 @ S3 ) )
       => ( ( inf_inf_set_int @ A3 @ S )
          = ( inf_inf_set_int @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2228_hoare__triple__def,axiom,
    ( hoare_8945653483474564448t_unit
    = ( ^ [P4: assn,C4: heap_T5738788834812785303t_unit,Q3: product_unit > assn] :
        ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
          ( ( rep_assn @ P4 @ ( produc7507926704131184380et_nat @ H6 @ As5 ) )
         => ? [R4: product_unit,H7: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] :
                  ( ( heap_T875086893843062177t_unit @ C4 @ H6 )
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H7 @ T3 ) ) ) )
              & ( rep_assn @ ( Q3 @ R4 ) @ ( produc7507926704131184380et_nat @ H7 @ ( hoare_new_addrs @ H6 @ As5 @ H7 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H6 ) )
                      & ~ ( member_nat @ A5 @ As5 ) ) )
                @ H6
                @ H7 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ).

% hoare_triple_def
thf(fact_2229_hoare__triple__def,axiom,
    ( hoare_hoare_triple_a
    = ( ^ [P4: assn,C4: heap_Time_Heap_a,Q3: a > assn] :
        ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
          ( ( rep_assn @ P4 @ ( produc7507926704131184380et_nat @ H6 @ As5 ) )
         => ? [R4: a,H7: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] :
                  ( ( heap_Time_execute_a @ C4 @ H6 )
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H7 @ T3 ) ) ) )
              & ( rep_assn @ ( Q3 @ R4 ) @ ( produc7507926704131184380et_nat @ H7 @ ( hoare_new_addrs @ H6 @ As5 @ H7 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H6 ) )
                      & ~ ( member_nat @ A5 @ As5 ) ) )
                @ H6
                @ H7 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ).

% hoare_triple_def
thf(fact_2230_hoare__tripleI,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn] :
      ( ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
         => ? [R5: product_unit,H8: heap_e7401611519738050253t_unit,T4: nat] :
              ( ( ( heap_T875086893843062177t_unit @ C @ H4 )
                = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R5 @ ( produc584006145561248582it_nat @ H8 @ T4 ) ) ) )
              & ( rep_assn @ ( Q @ R5 ) @ ( produc7507926704131184380et_nat @ H8 @ ( hoare_new_addrs @ H4 @ As4 @ H8 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H4 ) )
                      & ~ ( member_nat @ A5 @ As4 ) ) )
                @ H4
                @ H8 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H4 ) @ ( lim_Product_unit @ H8 ) ) ) )
     => ( hoare_8945653483474564448t_unit @ P @ C @ Q ) ) ).

% hoare_tripleI
thf(fact_2231_hoare__tripleI,axiom,
    ! [P: assn,C: heap_Time_Heap_a,Q: a > assn] :
      ( ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
         => ? [R5: a,H8: heap_e7401611519738050253t_unit,T4: nat] :
              ( ( ( heap_Time_execute_a @ C @ H4 )
                = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R5 @ ( produc584006145561248582it_nat @ H8 @ T4 ) ) ) )
              & ( rep_assn @ ( Q @ R5 ) @ ( produc7507926704131184380et_nat @ H8 @ ( hoare_new_addrs @ H4 @ As4 @ H8 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H4 ) )
                      & ~ ( member_nat @ A5 @ As4 ) ) )
                @ H4
                @ H8 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H4 ) @ ( lim_Product_unit @ H8 ) ) ) )
     => ( hoare_hoare_triple_a @ P @ C @ Q ) ) ).

% hoare_tripleI
thf(fact_2232_hoare__tripleE,axiom,
    ! [P: assn,C: heap_T5738788834812785303t_unit,Q: product_unit > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_8945653483474564448t_unit @ P @ C @ Q )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ~ ! [R2: product_unit,H5: heap_e7401611519738050253t_unit] :
              ( ? [T2: nat] :
                  ( ( heap_T875086893843062177t_unit @ C @ H )
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ T2 ) ) ) )
             => ( ( rep_assn @ ( Q @ R2 ) @ ( produc7507926704131184380et_nat @ H5 @ ( hoare_new_addrs @ H @ As @ H5 ) ) )
               => ( ( relH
                    @ ( collect_nat
                      @ ^ [A5: nat] :
                          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H ) )
                          & ~ ( member_nat @ A5 @ As ) ) )
                    @ H
                    @ H5 )
                 => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ H ) @ ( lim_Product_unit @ H5 ) ) ) ) ) ) ) ).

% hoare_tripleE
thf(fact_2233_hoare__tripleE,axiom,
    ! [P: assn,C: heap_Time_Heap_a,Q: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P @ C @ Q )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ~ ! [R2: a,H5: heap_e7401611519738050253t_unit] :
              ( ? [T2: nat] :
                  ( ( heap_Time_execute_a @ C @ H )
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ T2 ) ) ) )
             => ( ( rep_assn @ ( Q @ R2 ) @ ( produc7507926704131184380et_nat @ H5 @ ( hoare_new_addrs @ H @ As @ H5 ) ) )
               => ( ( relH
                    @ ( collect_nat
                      @ ^ [A5: nat] :
                          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H ) )
                          & ~ ( member_nat @ A5 @ As ) ) )
                    @ H
                    @ H5 )
                 => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ H ) @ ( lim_Product_unit @ H5 ) ) ) ) ) ) ) ).

% hoare_tripleE
thf(fact_2234_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila1307241728879085301at_nat @ sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat
    @ ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le6428140832669894131at_nat @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2235_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila5207306265035627125at_nat @ sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat
    @ ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( ord_le7642048601412989811at_nat @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2236_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila6534579987270727413at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat
    @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le2604355607129572851at_nat @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2237_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila4918476307565957903at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat
    @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2238_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila2554085542299052326_set_o @ sup_sup_set_o @ bot_bot_set_o
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_eq_set_o @ Y4 @ X4 )
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_set_o @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2239_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila1667268886620078168et_nat @ sup_sup_set_nat @ bot_bot_set_nat
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X4 )
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2240_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila8603258263270017530r_assn @ sup_sup_assn @ bot_bot_assn
    @ ^ [X4: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ X4 )
    @ ^ [X4: assn,Y4: assn] : ( ord_less_assn @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2241_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila6712789903965657268et_int @ sup_sup_set_int @ bot_bot_set_int
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ X4 )
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_set_int @ Y4 @ X4 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_2242_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ K2 @ I )
                 => ( P @ I ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_2243_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_2244_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_2245_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_2246_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N: nat] :
          ( ( ord_less_nat @ M3 @ N )
          | ( M3 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_2247_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_2248_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
          & ( M3 != N ) ) ) ) ).

% nat_less_le
thf(fact_2249_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member_set_nat @ X3 @ A3 )
     => ( ord_le6893508408891458716et_nat @ A3 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_2250_subset__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A3 )
     => ( ord_le3146513528884898305at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_2251_subset__emptyI,axiom,
    ! [A3: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A3 )
     => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_2252_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A3 )
     => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_2253_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A3 )
     => ( ord_less_eq_set_int @ A3 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_2254_psubsetI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( A3 != B3 )
       => ( ord_less_set_int @ A3 @ B3 ) ) ) ).

% psubsetI
thf(fact_2255_less__by__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( A3 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ).

% less_by_empty
thf(fact_2256_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
      ( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
     => ( ( Z
          = ( F @ A @ B ) )
        = ( ( A = Z )
          & ( B = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
thf(fact_2257_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
      ( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
     => ( ( ( F @ A @ B )
          = Z )
        = ( ( A = Z )
          & ( B = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
thf(fact_2258_not__psubset__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_2259_not__psubset__empty,axiom,
    ! [A3: set_o] :
      ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_2260_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_2261_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_2262_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2263_subset__psubset__trans,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C3 )
       => ( ord_less_set_int @ A3 @ C3 ) ) ) ).

% subset_psubset_trans
thf(fact_2264_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2265_psubset__subset__trans,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C3 )
       => ( ord_less_set_int @ A3 @ C3 ) ) ) ).

% psubset_subset_trans
thf(fact_2266_psubset__imp__subset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ( ord_less_eq_set_int @ A3 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_2267_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_2268_psubsetE,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B3 )
         => ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ).

% psubsetE
thf(fact_2269_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B5: assn] :
          ( ( ord_less_eq_assn @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% less_assn_def
thf(fact_2270_ssubst__Pair__rhs,axiom,
    ! [R3: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2271_ssubst__Pair__rhs,axiom,
    ! [R3: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R: set_Pr3286484037609594932et_nat,S4: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2272_ssubst__Pair__rhs,axiom,
    ! [R3: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R: set_Pr8536935166611901872et_nat,S4: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2273_ssubst__Pair__rhs,axiom,
    ! [R3: produc8551481072490612790e_term > option6357759511663192854e_term,S2: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S4: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2274_ssubst__Pair__rhs,axiom,
    ! [R3: int > option6357759511663192854e_term,S2: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S4: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2275_ssubst__Pair__rhs,axiom,
    ! [R3: a,S2: produc6653097349344004940it_nat,R: set_Pr7098220151150636591it_nat,S4: produc6653097349344004940it_nat] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ R3 @ S2 ) @ R )
     => ( ( S4 = S2 )
       => ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2276_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_2277_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_2278_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_2279_less__not__refl3,axiom,
    ! [S2: nat,T5: nat] :
      ( ( ord_less_nat @ S2 @ T5 )
     => ( S2 != T5 ) ) ).

% less_not_refl3
thf(fact_2280_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_2281_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
             => ( P @ M4 ) )
         => ( P @ N5 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_2282_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ~ ( P @ N5 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ~ ( P @ M4 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_2283_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_2284_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_2285_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_2286_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_2287_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_2288_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_2289_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_2290_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z4 )
     => ( ( ord_le3146513528884898305at_nat @ Z4
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2291_prop__restrict,axiom,
    ! [X: $o,Z4: set_o,X6: set_o,P: $o > $o] :
      ( ( member_o @ X @ Z4 )
     => ( ( ord_less_eq_set_o @ Z4
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2292_prop__restrict,axiom,
    ! [X: product_prod_int_int,Z4: set_Pr958786334691620121nt_int,X6: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ X @ Z4 )
     => ( ( ord_le2843351958646193337nt_int @ Z4
          @ ( collec213857154873943460nt_int
            @ ^ [X4: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2293_prop__restrict,axiom,
    ! [X: list_nat,Z4: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X @ Z4 )
     => ( ( ord_le6045566169113846134st_nat @ Z4
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2294_prop__restrict,axiom,
    ! [X: set_nat,Z4: set_set_nat,X6: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ Z4 )
     => ( ( ord_le6893508408891458716et_nat @ Z4
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2295_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X6: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2296_prop__restrict,axiom,
    ! [X: int,Z4: set_int,X6: set_int,P: int > $o] :
      ( ( member_int @ X @ Z4 )
     => ( ( ord_less_eq_set_int @ Z4
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ X6 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_2297_Collect__restrict,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2298_Collect__restrict,axiom,
    ! [X6: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2299_Collect__restrict,axiom,
    ! [X6: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2300_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2301_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2302_Collect__restrict,axiom,
    ! [X6: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2303_Collect__restrict,axiom,
    ! [X6: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ X6 )
            & ( P @ X4 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_2304_minf_I8_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ~ ( ord_less_eq_rat @ T5 @ X7 ) ) ).

% minf(8)
thf(fact_2305_minf_I8_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ~ ( ord_less_eq_num @ T5 @ X7 ) ) ).

% minf(8)
thf(fact_2306_minf_I8_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ~ ( ord_less_eq_nat @ T5 @ X7 ) ) ).

% minf(8)
thf(fact_2307_minf_I8_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ~ ( ord_less_eq_int @ T5 @ X7 ) ) ).

% minf(8)
thf(fact_2308_minf_I6_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( ord_less_eq_rat @ X7 @ T5 ) ) ).

% minf(6)
thf(fact_2309_minf_I6_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ( ord_less_eq_num @ X7 @ T5 ) ) ).

% minf(6)
thf(fact_2310_minf_I6_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( ord_less_eq_nat @ X7 @ T5 ) ) ).

% minf(6)
thf(fact_2311_minf_I6_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( ord_less_eq_int @ X7 @ T5 ) ) ).

% minf(6)
thf(fact_2312_pinf_I8_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( ord_less_eq_rat @ T5 @ X7 ) ) ).

% pinf(8)
thf(fact_2313_pinf_I8_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ( ord_less_eq_num @ T5 @ X7 ) ) ).

% pinf(8)
thf(fact_2314_pinf_I8_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( ord_less_eq_nat @ T5 @ X7 ) ) ).

% pinf(8)
thf(fact_2315_pinf_I8_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( ord_less_eq_int @ T5 @ X7 ) ) ).

% pinf(8)
thf(fact_2316_pinf_I6_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ~ ( ord_less_eq_rat @ X7 @ T5 ) ) ).

% pinf(6)
thf(fact_2317_pinf_I6_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ~ ( ord_less_eq_num @ X7 @ T5 ) ) ).

% pinf(6)
thf(fact_2318_pinf_I6_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ~ ( ord_less_eq_nat @ X7 @ T5 ) ) ).

% pinf(6)
thf(fact_2319_pinf_I6_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ~ ( ord_less_eq_int @ X7 @ T5 ) ) ).

% pinf(6)
thf(fact_2320_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2321_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
      = ( ord_less_num @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2322_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2323_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2324_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B )
              & ! [X7: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X7 )
                    & ( ord_less_nat @ X7 @ C2 ) )
                 => ( P @ X7 ) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D4 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D4 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_2325_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: int] :
              ( ( ord_less_eq_int @ A @ C2 )
              & ( ord_less_eq_int @ C2 @ B )
              & ! [X7: int] :
                  ( ( ( ord_less_eq_int @ A @ X7 )
                    & ( ord_less_int @ X7 @ C2 ) )
                 => ( P @ X7 ) )
              & ! [D4: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D4 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_int @ D4 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_2326_bijective__Empty,axiom,
    bijective_nat_nat @ bot_bo2099793752762293965at_nat ).

% bijective_Empty
thf(fact_2327_the__dflt__None__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ( ( dflt_N6592383573632408824at_nat @ S )
        = ( some_s147305329494351046at_nat @ S ) ) ) ).

% the_dflt_None_nonempty
thf(fact_2328_the__dflt__None__nonempty,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ( ( dflt_None_set_o @ S )
        = ( some_set_o @ S ) ) ) ).

% the_dflt_None_nonempty
thf(fact_2329_the__dflt__None__nonempty,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( ( dflt_None_set_nat @ S )
        = ( some_set_nat @ S ) ) ) ).

% the_dflt_None_nonempty
thf(fact_2330_the__dflt__None__nonempty,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ( ( dflt_None_set_int @ S )
        = ( some_set_int @ S ) ) ) ).

% the_dflt_None_nonempty
thf(fact_2331_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) )
               => ? [As13: set_nat,As23: set_nat] :
                    ( ( As4
                      = ( sup_sup_set_nat @ As13 @ As23 ) )
                    & ( ( inf_inf_set_nat @ As13 @ As23 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As13 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_2332_psubsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ B3 )
     => ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2333_psubsetD,axiom,
    ! [A3: set_o,B3: set_o,C: $o] :
      ( ( ord_less_set_o @ A3 @ B3 )
     => ( ( member_o @ C @ A3 )
       => ( member_o @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2334_psubsetD,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B3 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2335_psubsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2336_psubsetD,axiom,
    ! [A3: set_int,B3: set_int,C: int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2337_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_2338_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_o_o
          @ ^ [X4: $o] : ( member_o @ X4 @ A6 )
          @ ^ [X4: $o] : ( member_o @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_2339_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
          @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_2340_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_2341_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A6 )
          @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_2342_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_2343_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_2344_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_2345_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_2346_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2347_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2348_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2349_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2350_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2351_verit__comp__simplify1_I1_J,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_2352_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_2353_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_2354_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_2355_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_2356_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X7: rat] :
            ( ( ord_less_rat @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_2357_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X7: num] :
            ( ( ord_less_num @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_2358_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_2359_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X7: int] :
            ( ( ord_less_int @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_2360_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X7: rat] :
            ( ( ord_less_rat @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_2361_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X7: num] :
            ( ( ord_less_num @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_2362_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_2363_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X7: int] :
            ( ( ord_less_int @ Z2 @ X7 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_2364_pinf_I3_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(3)
thf(fact_2365_pinf_I3_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(3)
thf(fact_2366_pinf_I3_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(3)
thf(fact_2367_pinf_I3_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(3)
thf(fact_2368_pinf_I4_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(4)
thf(fact_2369_pinf_I4_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(4)
thf(fact_2370_pinf_I4_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(4)
thf(fact_2371_pinf_I4_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( X7 != T5 ) ) ).

% pinf(4)
thf(fact_2372_pinf_I5_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ~ ( ord_less_rat @ X7 @ T5 ) ) ).

% pinf(5)
thf(fact_2373_pinf_I5_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ~ ( ord_less_num @ X7 @ T5 ) ) ).

% pinf(5)
thf(fact_2374_pinf_I5_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ~ ( ord_less_nat @ X7 @ T5 ) ) ).

% pinf(5)
thf(fact_2375_pinf_I5_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ~ ( ord_less_int @ X7 @ T5 ) ) ).

% pinf(5)
thf(fact_2376_pinf_I7_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( ord_less_rat @ T5 @ X7 ) ) ).

% pinf(7)
thf(fact_2377_pinf_I7_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ Z2 @ X7 )
     => ( ord_less_num @ T5 @ X7 ) ) ).

% pinf(7)
thf(fact_2378_pinf_I7_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( ord_less_nat @ T5 @ X7 ) ) ).

% pinf(7)
thf(fact_2379_pinf_I7_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( ord_less_int @ T5 @ X7 ) ) ).

% pinf(7)
thf(fact_2380_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X7: rat] :
            ( ( ord_less_rat @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_2381_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X7: num] :
            ( ( ord_less_num @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_2382_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_2383_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X7: int] :
            ( ( ord_less_int @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                & ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                & ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(1)
thf(fact_2384_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q4: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X7: rat] :
            ( ( ord_less_rat @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_2385_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q4: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X7: num] :
            ( ( ord_less_num @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_2386_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X7: nat] :
            ( ( ord_less_nat @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_2387_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X7: int] :
            ( ( ord_less_int @ X7 @ Z2 )
           => ( ( ( P @ X7 )
                | ( Q @ X7 ) )
              = ( ( P6 @ X7 )
                | ( Q4 @ X7 ) ) ) ) ) ) ).

% minf(2)
thf(fact_2388_minf_I3_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(3)
thf(fact_2389_minf_I3_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(3)
thf(fact_2390_minf_I3_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(3)
thf(fact_2391_minf_I3_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(3)
thf(fact_2392_minf_I4_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(4)
thf(fact_2393_minf_I4_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(4)
thf(fact_2394_minf_I4_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(4)
thf(fact_2395_minf_I4_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( X7 != T5 ) ) ).

% minf(4)
thf(fact_2396_minf_I5_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( ord_less_rat @ X7 @ T5 ) ) ).

% minf(5)
thf(fact_2397_minf_I5_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ( ord_less_num @ X7 @ T5 ) ) ).

% minf(5)
thf(fact_2398_minf_I5_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( ord_less_nat @ X7 @ T5 ) ) ).

% minf(5)
thf(fact_2399_minf_I5_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( ord_less_int @ X7 @ T5 ) ) ).

% minf(5)
thf(fact_2400_minf_I7_J,axiom,
    ! [T5: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ~ ( ord_less_rat @ T5 @ X7 ) ) ).

% minf(7)
thf(fact_2401_minf_I7_J,axiom,
    ! [T5: num] :
    ? [Z2: num] :
    ! [X7: num] :
      ( ( ord_less_num @ X7 @ Z2 )
     => ~ ( ord_less_num @ T5 @ X7 ) ) ).

% minf(7)
thf(fact_2402_minf_I7_J,axiom,
    ! [T5: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ~ ( ord_less_nat @ T5 @ X7 ) ) ).

% minf(7)
thf(fact_2403_minf_I7_J,axiom,
    ! [T5: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ~ ( ord_less_int @ T5 @ X7 ) ) ).

% minf(7)
thf(fact_2404_bijective__def,axiom,
    ( bijective_nat_nat
    = ( ^ [R6: set_Pr1261947904930325089at_nat] :
          ( ! [X4: nat,Y4: nat,Z6: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R6 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: nat,Y4: nat,Z6: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z6 ) @ R6 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2405_bijective__def,axiom,
    ( biject2615096655818420098et_nat
    = ( ^ [R6: set_Pr3286484037609594932et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat,Z6: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R6 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z6: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z6 ) @ R6 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2406_bijective__def,axiom,
    ( biject1468766312547416318et_nat
    = ( ^ [R6: set_Pr8536935166611901872et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat,Z6: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R6 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z6: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z6 ) @ R6 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2407_bijective__def,axiom,
    ( biject383251550997737151nt_int
    = ( ^ [R6: set_Pr9222295170931077689nt_int] :
          ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int,Z6: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) @ R6 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: produc8551481072490612790e_term > option6357759511663192854e_term,Z6: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Z6 ) @ R6 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2408_bijective__def,axiom,
    ( biject576505603616484041nt_int
    = ( ^ [R6: set_Pr1872883991513573699nt_int] :
          ( ! [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int,Z6: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) @ R6 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: int > option6357759511663192854e_term,Y4: int > option6357759511663192854e_term,Z6: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Z6 ) @ R6 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2409_bijective__def,axiom,
    ( biject8592306251674886141it_nat
    = ( ^ [R6: set_Pr7098220151150636591it_nat] :
          ( ! [X4: a,Y4: produc6653097349344004940it_nat,Z6: produc6653097349344004940it_nat] :
              ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R6 )
                & ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Z6 ) @ R6 ) )
             => ( Y4 = Z6 ) )
          & ! [X4: a,Y4: a,Z6: produc6653097349344004940it_nat] :
              ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Z6 ) @ R6 )
                & ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ Y4 @ Z6 ) @ R6 ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_2410_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( Y
                  = ( ? [As12: set_nat,As22: set_nat] :
                        ( ( As4
                          = ( sup_sup_set_nat @ As12 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As12 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As12 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_2411_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) )
               => ~ ? [As1: set_nat,As2: set_nat] :
                      ( ( As4
                        = ( sup_sup_set_nat @ As1 @ As2 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As2 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H4 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As2 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_2412_the__dflt__None__set,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( the_de3668382347305825784at_nat @ bot_bo2099793752762293965at_nat @ ( dflt_N6592383573632408824at_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_2413_the__dflt__None__set,axiom,
    ! [X: set_o] :
      ( ( the_default_set_o @ bot_bot_set_o @ ( dflt_None_set_o @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_2414_the__dflt__None__set,axiom,
    ! [X: set_nat] :
      ( ( the_default_set_nat @ bot_bot_set_nat @ ( dflt_None_set_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_2415_the__dflt__None__set,axiom,
    ! [X: set_int] :
      ( ( the_default_set_int @ bot_bot_set_int @ ( dflt_None_set_int @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_2416_accp__subset,axiom,
    ! [R1: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o,R22: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o] :
      ( ( ord_le3355651325779393058_nat_o @ R1 @ R22 )
     => ( ord_le729326519192465773_nat_o @ ( accp_P5801069581201407417et_nat @ R22 ) @ ( accp_P5801069581201407417et_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2417_accp__subset,axiom,
    ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
      ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
     => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).

% accp_subset
thf(fact_2418_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2419_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_2420_accp__subset,axiom,
    ! [R1: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,R22: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o] :
      ( ( ord_le7226304311989208114_nat_o @ R1 @ R22 )
     => ( ord_le1658592502415435381_nat_o @ ( accp_P1862375125659990705et_nat @ R22 ) @ ( accp_P1862375125659990705et_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_2421_accp__subset__induct,axiom,
    ! [D3: produc3658429121746597890et_nat > $o,R: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
      ( ( ord_le729326519192465773_nat_o @ D3 @ ( accp_P5801069581201407417et_nat @ R ) )
     => ( ! [X3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
            ( ( D3 @ X3 )
           => ( ( R @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: produc3658429121746597890et_nat] :
                ( ( D3 @ X3 )
               => ( ! [Z5: produc3658429121746597890et_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2422_accp__subset__induct,axiom,
    ! [D3: product_prod_num_num > $o,R: product_prod_num_num > product_prod_num_num > $o,X: product_prod_num_num,P: product_prod_num_num > $o] :
      ( ( ord_le2239182809043710856_num_o @ D3 @ ( accp_P3113834385874906142um_num @ R ) )
     => ( ! [X3: product_prod_num_num,Z2: product_prod_num_num] :
            ( ( D3 @ X3 )
           => ( ( R @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: product_prod_num_num] :
                ( ( D3 @ X3 )
               => ( ! [Z5: product_prod_num_num] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2423_accp__subset__induct,axiom,
    ! [D3: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D3 @ ( accp_P4275260045618599050at_nat @ R ) )
     => ( ! [X3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( ( D3 @ X3 )
           => ( ( R @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( D3 @ X3 )
               => ( ! [Z5: product_prod_nat_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2424_accp__subset__induct,axiom,
    ! [D3: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D3 @ ( accp_P1096762738010456898nt_int @ R ) )
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( ( D3 @ X3 )
           => ( ( R @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: product_prod_int_int] :
                ( ( D3 @ X3 )
               => ( ! [Z5: product_prod_int_int] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2425_accp__subset__induct,axiom,
    ! [D3: produc2732055786443039994et_nat > $o,R: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat,P: produc2732055786443039994et_nat > $o] :
      ( ( ord_le1658592502415435381_nat_o @ D3 @ ( accp_P1862375125659990705et_nat @ R ) )
     => ( ! [X3: produc2732055786443039994et_nat,Z2: produc2732055786443039994et_nat] :
            ( ( D3 @ X3 )
           => ( ( R @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: produc2732055786443039994et_nat] :
                ( ( D3 @ X3 )
               => ( ! [Z5: produc2732055786443039994et_nat] :
                      ( ( R @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_2426_bot_Oordering__top__axioms,axiom,
    ( orderi2172309028950807442at_nat
    @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y4 @ X4 )
    @ bot_bo2099793752762293965at_nat ) ).

% bot.ordering_top_axioms
thf(fact_2427_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_o
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_eq_set_o @ Y4 @ X4 )
    @ ^ [X4: set_o,Y4: set_o] : ( ord_less_set_o @ Y4 @ X4 )
    @ bot_bot_set_o ) ).

% bot.ordering_top_axioms
thf(fact_2428_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_nat
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X4 )
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X4 )
    @ bot_bot_set_nat ) ).

% bot.ordering_top_axioms
thf(fact_2429_bot_Oordering__top__axioms,axiom,
    ( ordering_top_assn
    @ ^ [X4: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ X4 )
    @ ^ [X4: assn,Y4: assn] : ( ord_less_assn @ Y4 @ X4 )
    @ bot_bot_assn ) ).

% bot.ordering_top_axioms
thf(fact_2430_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_int
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ X4 )
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_set_int @ Y4 @ X4 )
    @ bot_bot_set_int ) ).

% bot.ordering_top_axioms
thf(fact_2431_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_2432_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_2433_hoare__triple__effect,axiom,
    ! [P: assn,C: heap_Time_Heap_a,Q: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P @ C @ Q )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ? [H5: heap_e7401611519738050253t_unit,R2: a,T2: nat] :
            ( ( heap_Time_effect_a @ C @ H @ H5 @ R2 @ T2 )
            & ( rep_assn @ ( Q @ R2 ) @ ( produc7507926704131184380et_nat @ H5 @ ( hoare_new_addrs @ H @ As @ H5 ) ) ) ) ) ) ).

% hoare_triple_effect
thf(fact_2434_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic5672175589177086241at_nat @ sup_su718114333110466843at_nat
    @ ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le6428140832669894131at_nat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2435_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic5174734335190411425at_nat @ sup_su3035147773818789531at_nat
    @ ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( ord_le7642048601412989811at_nat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2436_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic6976360593897529633at_nat @ sup_su5525570899277871387at_nat
    @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ X4 )
    @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le2604355607129572851at_nat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2437_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic3109210760196336428et_nat @ sup_sup_set_nat
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X4 )
    @ ^ [X4: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2438_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic5623431474481994958t_assn @ sup_sup_assn
    @ ^ [X4: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ X4 )
    @ ^ [X4: assn,Y4: assn] : ( ord_less_assn @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2439_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic8154731777541915528et_int @ sup_sup_set_int
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ X4 )
    @ ^ [X4: set_int,Y4: set_int] : ( ord_less_set_int @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2440_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic5374021519246970238et_rat @ sup_sup_rat
    @ ^ [X4: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ X4 )
    @ ^ [X4: rat,Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2441_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic6009151579333465974et_nat @ sup_sup_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2442_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic6006661108824415698et_int @ sup_sup_int
    @ ^ [X4: int,Y4: int] : ( ord_less_eq_int @ Y4 @ X4 )
    @ ^ [X4: int,Y4: int] : ( ord_less_int @ Y4 @ X4 ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_2443_wand__assnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,R: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( inf_inf_set_nat @ As @ As6 )
              = bot_bot_set_nat )
           => ( ( relH @ As @ H @ H5 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
               => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H5 @ As6 ) )
                 => ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q @ R ) @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% wand_assnI
thf(fact_2444_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_2445_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( ( times_times_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_star
thf(fact_2446_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          | B ) ) ) ).

% merge_pure_or
thf(fact_2447_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_2448_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_2449_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_and
thf(fact_2450_in__range__empty,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_2451_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ H )
      = ( ( rep_assn @ P @ H )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_2452_in__range__dist__union,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat,As3: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As @ As3 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H @ As3 ) ) ) ) ).

% in_range_dist_union
thf(fact_2453_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
thf(fact_2454_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( A != Top )
        = ( Less @ A @ Top ) ) ) ).

% ordering_top.not_eq_extremum
thf(fact_2455_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
        = ( A = Top ) ) ) ).

% ordering_top.extremum_unique
thf(fact_2456_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ~ ( Less @ Top @ A ) ) ).

% ordering_top.extremum_strict
thf(fact_2457_ordering__top_Oextremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( Less_eq @ A @ Top ) ) ).

% ordering_top.extremum
thf(fact_2458_models__in__range,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H )
     => ( in_range @ H ) ) ).

% models_in_range
thf(fact_2459_relH__in__rangeI_I2_J,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H3 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).

% relH_in_rangeI(2)
thf(fact_2460_relH__in__rangeI_I1_J,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H3 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ).

% relH_in_rangeI(1)
thf(fact_2461_relH__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( relH @ As @ H @ H ) ) ).

% relH_refl
thf(fact_2462_in__range__subset,axiom,
    ! [As: set_nat,As3: set_nat,H: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As @ As3 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As3 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% in_range_subset
thf(fact_2463_effectI,axiom,
    ! [C: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,R3: product_unit,H3: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_T875086893843062177t_unit @ C @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R3 @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( heap_T6553295506729943825t_unit @ C @ H @ H3 @ R3 @ N2 ) ) ).

% effectI
thf(fact_2464_effectI,axiom,
    ! [C: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,R3: a,H3: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_Time_execute_a @ C @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R3 @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( heap_Time_effect_a @ C @ H @ H3 @ R3 @ N2 ) ) ).

% effectI
thf(fact_2465_effect__def,axiom,
    ( heap_T6553295506729943825t_unit
    = ( ^ [C4: heap_T5738788834812785303t_unit,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R4: product_unit,N: nat] :
          ( ( heap_T875086893843062177t_unit @ C4 @ H6 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H7 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_2466_effect__def,axiom,
    ( heap_Time_effect_a
    = ( ^ [C4: heap_Time_Heap_a,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R4: a,N: nat] :
          ( ( heap_Time_execute_a @ C4 @ H6 )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H7 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_2467_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ! [X3: nat] :
                ( ( member_nat @ X3 @ As4 )
               => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H4 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_2468_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ~ ! [X7: nat] :
                  ( ( member_nat @ X7 @ As4 )
                 => ( ord_less_nat @ X7 @ ( lim_Product_unit @ H4 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_2469_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( Y
              = ( ~ ! [X4: nat] :
                      ( ( member_nat @ X4 @ As4 )
                     => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_2470_in__range_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ As )
           => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H ) ) ) ) ) ).

% in_range.simps
thf(fact_2471_Heap__eqI,axiom,
    ! [F: heap_Time_Heap_a,G: heap_Time_Heap_a] :
      ( ! [H4: heap_e7401611519738050253t_unit] :
          ( ( heap_Time_execute_a @ F @ H4 )
          = ( heap_Time_execute_a @ G @ H4 ) )
     => ( F = G ) ) ).

% Heap_eqI
thf(fact_2472_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2473_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5111740090977526247t_unit @ inf_inf_Product_unit @ ord_le3221252021190050221t_unit @ ord_le361264281704409273t_unit ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2474_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6529551498584149819at_nat @ inf_in2572325071724192079at_nat @ ord_le3146513528884898305at_nat @ ord_le7866589430770878221at_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2475_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn @ inf_inf_assn @ ord_less_eq_assn @ ord_less_assn ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2476_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8154731777541915528et_int @ inf_inf_set_int @ ord_less_eq_set_int @ ord_less_set_int ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2477_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat @ inf_inf_rat @ ord_less_eq_rat @ ord_less_rat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2478_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2479_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int @ inf_inf_int @ ord_less_eq_int @ ord_less_int ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2480_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc3260487557148687353it_nat,X: produc3260487557148687353it_nat] :
      ( ( the_de542837501814275618it_nat @ Uu @ ( some_P7913643980934408916it_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_2481_the__default_Osimps_I1_J,axiom,
    ! [Uu: num,X: num] :
      ( ( the_default_num @ Uu @ ( some_num @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_2482_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc8664842809031399944it_nat,X: produc8664842809031399944it_nat] :
      ( ( the_de2487931475039285041it_nat @ Uu @ ( some_P1914260805536162275it_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_2483_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
              & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As4 @ As6 )
                      = bot_bot_set_nat )
                    & ( relH @ As4 @ H4 @ H5 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As4 @ As6 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_2484_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                & ! [H8: heap_e7401611519738050253t_unit,As7: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As4 @ As7 )
                        = bot_bot_set_nat )
                      & ( relH @ As4 @ H4 @ H8 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As4 ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H8 @ As7 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As4 @ As7 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_2485_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 )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As4 @ As8 )
                            = bot_bot_set_nat )
                          & ( relH @ As4 @ H4 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As4 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As4 @ As8 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_2486_wand__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( wand_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
        & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As @ As8 )
                = bot_bot_set_nat )
              & ( relH @ As @ H @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
              & ( P @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
           => ( Q @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As8 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_2487_in__measure,axiom,
    ! [X: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_2488_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                  & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As4 @ As6 )
                          = bot_bot_set_nat )
                        & ( relH @ As4 @ H4 @ H5 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As4 @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_2489_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                    & ! [H8: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As4 @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As4 @ H4 @ H8 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As4 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H8 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As4 @ As7 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_2490_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 ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As4 @ As8 )
                            = bot_bot_set_nat )
                          & ( relH @ As4 @ H4 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As4 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As4 @ As8 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_2491_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
               => ! [X3: nat] :
                    ( ( member_nat @ X3 @ As4 )
                   => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_2492_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
               => ~ ! [X7: nat] :
                      ( ( member_nat @ X7 @ As4 )
                     => ( ord_less_nat @ X7 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_2493_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( Y
                  = ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ As4 )
                       => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H4 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_2494_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P4: assn,Q3: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P4 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% wand_assn_def
thf(fact_2495_sup__option__def,axiom,
    ( sup_su1419176701018045153at_nat
    = ( ^ [X4: option2158460348403737357at_nat,Y4: option2158460348403737357at_nat] :
          ( case_o7467001542679133189at_nat @ Y4
          @ ^ [X8: set_Pr8693737435421807431at_nat] :
              ( case_o7467001542679133189at_nat @ X4
              @ ^ [Z6: set_Pr8693737435421807431at_nat] : ( some_s579751585967276588at_nat @ ( sup_su718114333110466843at_nat @ X8 @ Z6 ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_2496_sup__option__def,axiom,
    ( sup_su5873096292857799009at_nat
    = ( ^ [X4: option2498585697089621389at_nat,Y4: option2498585697089621389at_nat] :
          ( case_o8757249484648823045at_nat @ Y4
          @ ^ [X8: set_Pr8551490117392284871at_nat] :
              ( case_o8757249484648823045at_nat @ X4
              @ ^ [Z6: set_Pr8551490117392284871at_nat] : ( some_s287724117700012716at_nat @ ( sup_su3035147773818789531at_nat @ X8 @ Z6 ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_2497_sup__option__def,axiom,
    ( sup_su2273273666271716065at_nat
    = ( ^ [X4: option1583680563626158861at_nat,Y4: option1583680563626158861at_nat] :
          ( case_o9080059180780454917at_nat @ Y4
          @ ^ [X8: set_Pr4329608150637261639at_nat] :
              ( case_o9080059180780454917at_nat @ X4
              @ ^ [Z6: set_Pr4329608150637261639at_nat] : ( some_s5890477192898017836at_nat @ ( sup_su5525570899277871387at_nat @ X8 @ Z6 ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_2498_sup__option__def,axiom,
    ( sup_su3598758113090618626et_nat
    = ( ^ [X4: option_set_nat,Y4: option_set_nat] :
          ( case_o4054078431260844265et_nat @ Y4
          @ ^ [X8: set_nat] :
              ( case_o4054078431260844265et_nat @ X4
              @ ^ [Z6: set_nat] : ( some_set_nat @ ( sup_sup_set_nat @ X8 @ Z6 ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_2499_sup__option__def,axiom,
    ( sup_sup_option_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_o7429725398727453821at_nat @ Y4
          @ ^ [X8: nat] :
              ( case_o7429725398727453821at_nat @ X4
              @ ^ [Z6: nat] : ( some_nat @ ( sup_sup_nat @ X8 @ Z6 ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_2500_dflt__None__set__def,axiom,
    ( dflt_N6592383573632408824at_nat
    = ( ^ [S5: set_Pr1261947904930325089at_nat] : ( if_opt7704869406773131885at_nat @ ( S5 = bot_bo2099793752762293965at_nat ) @ none_s625347054029921090at_nat @ ( some_s147305329494351046at_nat @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2501_dflt__None__set__def,axiom,
    ( dflt_None_set_o
    = ( ^ [S5: set_o] : ( if_option_set_o @ ( S5 = bot_bot_set_o ) @ none_set_o @ ( some_set_o @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2502_dflt__None__set__def,axiom,
    ( dflt_None_set_nat
    = ( ^ [S5: set_nat] : ( if_option_set_nat @ ( S5 = bot_bot_set_nat ) @ none_set_nat @ ( some_set_nat @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2503_dflt__None__set__def,axiom,
    ( dflt_None_set_int
    = ( ^ [S5: set_int] : ( if_option_set_int @ ( S5 = bot_bot_set_int ) @ none_set_int @ ( some_set_int @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2504_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P4: assn,Q3: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P4 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% times_assn_def
thf(fact_2505_Set_Ois__empty__def,axiom,
    ( is_emp1662574758705540307at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] : A6 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
thf(fact_2506_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : A6 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_2507_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : A6 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_2508_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A6: set_int] : A6 = bot_bot_set_int ) ) ).

% Set.is_empty_def
thf(fact_2509_hoare__triple__success,axiom,
    ! [P: assn,C: heap_Time_Heap_a,Q: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P @ C @ Q )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ( heap_Time_success_a @ C @ H ) ) ) ).

% hoare_triple_success
thf(fact_2510_subset__Collect__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2511_subset__Collect__iff,axiom,
    ! [B3: set_o,A3: set_o,P: $o > $o] :
      ( ( ord_less_eq_set_o @ B3 @ A3 )
     => ( ( ord_less_eq_set_o @ B3
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: $o] :
              ( ( member_o @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2512_subset__Collect__iff,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
     => ( ( ord_le2843351958646193337nt_int @ B3
          @ ( collec213857154873943460nt_int
            @ ^ [X4: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2513_subset__Collect__iff,axiom,
    ! [B3: set_list_nat,A3: set_list_nat,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B3 @ A3 )
     => ( ( ord_le6045566169113846134st_nat @ B3
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: list_nat] :
              ( ( member_list_nat @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2514_subset__Collect__iff,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ B3
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2515_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( ord_less_eq_set_nat @ B3
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2516_subset__Collect__iff,axiom,
    ! [B3: set_int,A3: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( ord_less_eq_set_int @ B3
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ B3 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_2517_subset__CollectI,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2518_subset__CollectI,axiom,
    ! [B3: set_o,A3: set_o,Q: $o > $o,P: $o > $o] :
      ( ( ord_less_eq_set_o @ B3 @ A3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2519_subset__CollectI,axiom,
    ! [B3: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,Q: product_prod_int_int > $o,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ B3 @ A3 )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le2843351958646193337nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X4: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collec213857154873943460nt_int
            @ ^ [X4: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2520_subset__CollectI,axiom,
    ! [B3: set_list_nat,A3: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B3 @ A3 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2521_subset__CollectI,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2522_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2523_subset__CollectI,axiom,
    ! [B3: set_int,A3: set_int,Q: int > $o,P: int > $o] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ B3 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ B3 )
                & ( Q @ X4 ) ) )
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_2524_conj__subset__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ A3
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_le2843351958646193337nt_int @ A3 @ ( collec213857154873943460nt_int @ P ) )
        & ( ord_le2843351958646193337nt_int @ A3 @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_2525_conj__subset__def,axiom,
    ! [A3: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A3
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ P ) )
        & ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_2526_conj__subset__def,axiom,
    ! [A3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A3
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ P ) )
        & ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_2527_conj__subset__def,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_2528_conj__subset__def,axiom,
    ! [A3: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A3
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_less_eq_set_int @ A3 @ ( collect_int @ P ) )
        & ( ord_less_eq_set_int @ A3 @ ( collect_int @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_2529_not__None__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( X != none_P2651198173097904984it_nat )
      = ( ? [Y4: produc3260487557148687353it_nat] :
            ( X
            = ( some_P7913643980934408916it_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_2530_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y4: num] :
            ( X
            = ( some_num @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_2531_not__None__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( X != none_P9117596204409417319it_nat )
      = ( ? [Y4: produc8664842809031399944it_nat] :
            ( X
            = ( some_P1914260805536162275it_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_2532_not__Some__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( ! [Y4: produc3260487557148687353it_nat] :
            ( X
           != ( some_P7913643980934408916it_nat @ Y4 ) ) )
      = ( X = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq
thf(fact_2533_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y4: num] :
            ( X
           != ( some_num @ Y4 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_2534_not__Some__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( ! [Y4: produc8664842809031399944it_nat] :
            ( X
           != ( some_P1914260805536162275it_nat @ Y4 ) ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
thf(fact_2535_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_2536_less__eq__option__None__code,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None_code
thf(fact_2537_less__option__None,axiom,
    ! [X: option_num] :
      ~ ( ord_less_option_num @ X @ none_num ) ).

% less_option_None
thf(fact_2538_less__eq__option__Some__None,axiom,
    ! [X: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_2539_less__option__None__Some__code,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some_code
thf(fact_2540_the__dflt__None__empty,axiom,
    ( ( dflt_N6592383573632408824at_nat @ bot_bo2099793752762293965at_nat )
    = none_s625347054029921090at_nat ) ).

% the_dflt_None_empty
thf(fact_2541_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_o @ bot_bot_set_o )
    = none_set_o ) ).

% the_dflt_None_empty
thf(fact_2542_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_nat @ bot_bot_set_nat )
    = none_set_nat ) ).

% the_dflt_None_empty
thf(fact_2543_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_int @ bot_bot_set_int )
    = none_set_int ) ).

% the_dflt_None_empty
thf(fact_2544_option_Osimps_I4_J,axiom,
    ! [F1: int,F22: num > int] :
      ( ( case_option_int_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_2545_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: num > option_num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_2546_option_Osimps_I4_J,axiom,
    ! [F1: num,F22: num > num] :
      ( ( case_option_num_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_2547_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
thf(fact_2548_option_Ocase__distrib,axiom,
    ! [H: int > int,F1: int,F22: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2549_option_Ocase__distrib,axiom,
    ! [H: int > option_num,F1: int,F22: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2550_option_Ocase__distrib,axiom,
    ! [H: int > num,F1: int,F22: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2551_option_Ocase__distrib,axiom,
    ! [H: option_num > int,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2552_option_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2553_option_Ocase__distrib,axiom,
    ! [H: option_num > num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2554_option_Ocase__distrib,axiom,
    ! [H: num > int,F1: num,F22: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2555_option_Ocase__distrib,axiom,
    ! [H: num > option_num,F1: num,F22: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2556_option_Ocase__distrib,axiom,
    ! [H: num > num,F1: num,F22: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_2557_option_Odistinct_I1_J,axiom,
    ! [X2: produc3260487557148687353it_nat] :
      ( none_P2651198173097904984it_nat
     != ( some_P7913643980934408916it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_2558_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_2559_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( none_P9117596204409417319it_nat
     != ( some_P1914260805536162275it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_2560_option_OdiscI,axiom,
    ! [Option: option3562590408128118217it_nat,X2: produc3260487557148687353it_nat] :
      ( ( Option
        = ( some_P7913643980934408916it_nat @ X2 ) )
     => ( Option != none_P2651198173097904984it_nat ) ) ).

% option.discI
thf(fact_2561_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_2562_option_OdiscI,axiom,
    ! [Option: option8956607266484857688it_nat,X2: produc8664842809031399944it_nat] :
      ( ( Option
        = ( some_P1914260805536162275it_nat @ X2 ) )
     => ( Option != none_P9117596204409417319it_nat ) ) ).

% option.discI
thf(fact_2563_option_Oexhaust,axiom,
    ! [Y: option3562590408128118217it_nat] :
      ( ( Y != none_P2651198173097904984it_nat )
     => ~ ! [X22: produc3260487557148687353it_nat] :
            ( Y
           != ( some_P7913643980934408916it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_2564_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] :
            ( Y
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_2565_option_Oexhaust,axiom,
    ! [Y: option8956607266484857688it_nat] :
      ( ( Y != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] :
            ( Y
           != ( some_P1914260805536162275it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_2566_split__option__ex,axiom,
    ( ( ^ [P5: option3562590408128118217it_nat > $o] :
        ? [X5: option3562590408128118217it_nat] : ( P5 @ X5 ) )
    = ( ^ [P4: option3562590408128118217it_nat > $o] :
          ( ( P4 @ none_P2651198173097904984it_nat )
          | ? [X4: produc3260487557148687353it_nat] : ( P4 @ ( some_P7913643980934408916it_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_2567_split__option__ex,axiom,
    ( ( ^ [P5: option_num > $o] :
        ? [X5: option_num] : ( P5 @ X5 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_2568_split__option__ex,axiom,
    ( ( ^ [P5: option8956607266484857688it_nat > $o] :
        ? [X5: option8956607266484857688it_nat] : ( P5 @ X5 ) )
    = ( ^ [P4: option8956607266484857688it_nat > $o] :
          ( ( P4 @ none_P9117596204409417319it_nat )
          | ? [X4: produc8664842809031399944it_nat] : ( P4 @ ( some_P1914260805536162275it_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_2569_split__option__all,axiom,
    ( ( ^ [P5: option3562590408128118217it_nat > $o] :
        ! [X5: option3562590408128118217it_nat] : ( P5 @ X5 ) )
    = ( ^ [P4: option3562590408128118217it_nat > $o] :
          ( ( P4 @ none_P2651198173097904984it_nat )
          & ! [X4: produc3260487557148687353it_nat] : ( P4 @ ( some_P7913643980934408916it_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_2570_split__option__all,axiom,
    ( ( ^ [P5: option_num > $o] :
        ! [X5: option_num] : ( P5 @ X5 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_2571_split__option__all,axiom,
    ( ( ^ [P5: option8956607266484857688it_nat > $o] :
        ! [X5: option8956607266484857688it_nat] : ( P5 @ X5 ) )
    = ( ^ [P4: option8956607266484857688it_nat > $o] :
          ( ( P4 @ none_P9117596204409417319it_nat )
          & ! [X4: produc8664842809031399944it_nat] : ( P4 @ ( some_P1914260805536162275it_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_2572_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,P: option3562590408128118217it_nat > option3562590408128118217it_nat > $o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ A4 ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2573_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,P: option3562590408128118217it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: num] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ A4 ) )
             => ( ( Y
                  = ( some_num @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2574_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,P: option3562590408128118217it_nat > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc3260487557148687353it_nat,B4: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ A4 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2575_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option3562590408128118217it_nat > $o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: num,B4: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2576_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A4: num,B4: num] :
              ( ( X
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_num @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2577_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: num,B4: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_num @ A4 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2578_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option3562590408128118217it_nat > $o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A4 ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2579_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: num] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A4 ) )
             => ( ( Y
                  = ( some_num @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2580_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A4: produc8664842809031399944it_nat,B4: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A4 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B4 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2581_less__eq__option__None,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None
thf(fact_2582_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( ( ord_le6622620407824499402on_num @ X @ none_num )
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
thf(fact_2583_Abs__assn__eqI_I1_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( P @ H4 )
          = ( rep_assn @ Pr @ H4 ) )
     => ( ( abs_assn @ P )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_2584_Abs__assn__eqI_I2_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( P @ H4 )
          = ( rep_assn @ Pr @ H4 ) )
     => ( Pr
        = ( abs_assn @ P ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_2585_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: num > int,X2: num] :
      ( ( case_option_int_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_2586_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: num > option_num,X2: num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_2587_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: num > num,X2: num] :
      ( ( case_option_num_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_2588_the__default_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( the_default_num @ X @ none_num )
      = X ) ).

% the_default.simps(2)
thf(fact_2589_inf__option__def,axiom,
    ( inf_in7812914138253463912et_nat
    = ( ^ [X4: option_set_nat,Y4: option_set_nat] :
          ( case_o4054078431260844265et_nat @ none_set_nat
          @ ^ [Z6: set_nat] :
              ( case_o4054078431260844265et_nat @ none_set_nat
              @ ^ [Aa: set_nat] : ( some_set_nat @ ( inf_inf_set_nat @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_2590_inf__option__def,axiom,
    ( inf_inf_option_assn
    = ( ^ [X4: option_assn,Y4: option_assn] :
          ( case_o4484465799723439917n_assn @ none_assn
          @ ^ [Z6: assn] :
              ( case_o4484465799723439917n_assn @ none_assn
              @ ^ [Aa: assn] : ( some_assn @ ( inf_inf_assn @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_2591_inf__option__def,axiom,
    ( inf_inf_option_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_o7429725398727453821at_nat @ none_nat
          @ ^ [Z6: nat] :
              ( case_o7429725398727453821at_nat @ none_nat
              @ ^ [Aa: nat] : ( some_nat @ ( inf_inf_nat @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_2592_inf__option__def,axiom,
    ( inf_in5800606384722953409t_unit
    = ( ^ [X4: option_Product_unit,Y4: option_Product_unit] :
          ( case_o2985186191577459077t_unit @ none_Product_unit
          @ ^ [Z6: product_unit] :
              ( case_o2985186191577459077t_unit @ none_Product_unit
              @ ^ [Aa: product_unit] : ( some_Product_unit @ ( inf_inf_Product_unit @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_2593_inf__option__def,axiom,
    ( inf_in777885744494410645at_nat
    = ( ^ [X4: option8963830502488799655at_nat,Y4: option8963830502488799655at_nat] :
          ( case_o311359030874850053at_nat @ none_s625347054029921090at_nat
          @ ^ [Z6: set_Pr1261947904930325089at_nat] :
              ( case_o311359030874850053at_nat @ none_s625347054029921090at_nat
              @ ^ [Aa: set_Pr1261947904930325089at_nat] : ( some_s147305329494351046at_nat @ ( inf_in2572325071724192079at_nat @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_2594_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_2595_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( ( ord_less_option_num @ none_num @ X )
     => ? [Z2: num] :
          ( X
          = ( some_num @ Z2 ) ) ) ).

% less_option_None_is_Some
thf(fact_2596_less__option__None__Some,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some
thf(fact_2597_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P4: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P4 @ H6 )
              | ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).

% sup_assn_def
thf(fact_2598_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P4: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P4 @ H6 )
              & ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).

% inf_assn_def
thf(fact_2599_successE,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T584514906347983379t_unit @ F @ H )
     => ~ ! [R2: product_unit,H5: produc6653097349344004940it_nat] :
            ( ( heap_T875086893843062177t_unit @ F @ H )
           != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ H5 ) ) ) ) ).

% successE
thf(fact_2600_successE,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_success_a @ F @ H )
     => ~ ! [R2: a,H5: produc6653097349344004940it_nat] :
            ( ( heap_Time_execute_a @ F @ H )
           != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ H5 ) ) ) ) ).

% successE
thf(fact_2601_rel__of__empty,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( rel_of_nat_nat
        @ ^ [X4: nat] : none_nat
        @ P )
      = bot_bo2099793752762293965at_nat ) ).

% rel_of_empty
thf(fact_2602_set__to__map__empty,axiom,
    ( ( set_to_map_nat_nat @ bot_bo2099793752762293965at_nat )
    = ( ^ [X4: nat] : none_nat ) ) ).

% set_to_map_empty
thf(fact_2603_map__to__set__empty,axiom,
    ( ( map_to_set_nat_nat
      @ ^ [X4: nat] : none_nat )
    = bot_bo2099793752762293965at_nat ) ).

% map_to_set_empty
thf(fact_2604_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R: int > $o,F: int,G: num > int] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q5: num] :
              ( ( Q @ Q5 )
             => ( R @ ( G @ Q5 ) ) )
         => ( R @ ( case_option_int_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_2605_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R: option_num > $o,F: option_num,G: num > option_num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q5: num] :
              ( ( Q @ Q5 )
             => ( R @ ( G @ Q5 ) ) )
         => ( R @ ( case_o6005452278849405969um_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_2606_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R: num > $o,F: num,G: num > num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q5: num] :
              ( ( Q @ Q5 )
             => ( R @ ( G @ Q5 ) ) )
         => ( R @ ( case_option_num_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_2607_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B5: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B5 ) ) ) ) ).

% pure_assn_def
thf(fact_2608_set__to__map__empty__iff_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat] : none_nat )
        = ( set_to_map_nat_nat @ S ) )
      = ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(2)
thf(fact_2609_not__Some__eq2,axiom,
    ! [V: option5190343406534369742et_nat] :
      ( ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] :
            ( V
           != ( some_P750831030444334937et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) ) ) )
      = ( V = none_P4972525538344268765et_nat ) ) ).

% not_Some_eq2
thf(fact_2610_not__Some__eq2,axiom,
    ! [V: option2860828798490689354et_nat] :
      ( ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] :
            ( V
           != ( some_P1630309045189364437et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) ) ) )
      = ( V = none_P199884684680593241et_nat ) ) ).

% not_Some_eq2
thf(fact_2611_not__Some__eq2,axiom,
    ! [V: option7541221861074943443nt_int] :
      ( ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( V
           != ( some_P2355398578364412894nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y4 ) ) ) )
      = ( V = none_P1286213070022356066nt_int ) ) ).

% not_Some_eq2
thf(fact_2612_not__Some__eq2,axiom,
    ! [V: option4256020574406277085nt_int] :
      ( ( ! [X4: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( V
           != ( some_P7455497367792166888nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y4 ) ) ) )
      = ( V = none_P3773570700014501484nt_int ) ) ).

% not_Some_eq2
thf(fact_2613_not__Some__eq2,axiom,
    ! [V: option3562590408128118217it_nat] :
      ( ( ! [X4: a,Y4: produc6653097349344004940it_nat] :
            ( V
           != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) ) ) )
      = ( V = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq2
thf(fact_2614_not__Some__eq2,axiom,
    ! [V: option8956607266484857688it_nat] :
      ( ( ! [X4: product_unit,Y4: produc6653097349344004940it_nat] :
            ( V
           != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X4 @ Y4 ) ) ) )
      = ( V = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq2
thf(fact_2615_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,G: a > heap_Time_Heap_a] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = none_P2651198173097904984it_nat )
     => ( ( heap_Time_execute_a @ ( heap_Time_bind_a_a @ F @ G ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
thf(fact_2616_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
      = ( case_option_o_num @ $false
        @ ^ [Uu2: num] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_2617_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
      = ( case_option_o_num @ $true
        @ ^ [Uu2: num] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_2618_success__def,axiom,
    ( heap_Time_success_a
    = ( ^ [F3: heap_Time_Heap_a,H6: heap_e7401611519738050253t_unit] :
          ( ( heap_Time_execute_a @ F3 @ H6 )
         != none_P2651198173097904984it_nat ) ) ) ).

% success_def
thf(fact_2619_successI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ( ( heap_Time_execute_a @ F @ H )
       != none_P2651198173097904984it_nat )
     => ( heap_Time_success_a @ F @ H ) ) ).

% successI
thf(fact_2620_case__optionE,axiom,
    ! [P: $o,Q: produc3260487557148687353it_nat > $o,X: option3562590408128118217it_nat] :
      ( ( case_o3267218206291580230it_nat @ P @ Q @ X )
     => ( ( ( X = none_P2651198173097904984it_nat )
         => ~ P )
       => ~ ! [Y3: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_2621_case__optionE,axiom,
    ! [P: $o,Q: num > $o,X: option_num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( ( X = none_num )
         => ~ P )
       => ~ ! [Y3: num] :
              ( ( X
                = ( some_num @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_2622_case__optionE,axiom,
    ! [P: $o,Q: produc8664842809031399944it_nat > $o,X: option8956607266484857688it_nat] :
      ( ( case_o2686588417244861013it_nat @ P @ Q @ X )
     => ( ( ( X = none_P9117596204409417319it_nat )
         => ~ P )
       => ~ ! [Y3: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_2623_map__to__set__empty__iff_I1_J,axiom,
    ! [M: nat > option_nat] :
      ( ( ( map_to_set_nat_nat @ M )
        = bot_bo2099793752762293965at_nat )
      = ( M
        = ( ^ [X4: nat] : none_nat ) ) ) ).

% map_to_set_empty_iff(1)
thf(fact_2624_map__to__set__empty__iff_I2_J,axiom,
    ! [M: nat > option_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( map_to_set_nat_nat @ M ) )
      = ( M
        = ( ^ [X4: nat] : none_nat ) ) ) ).

% map_to_set_empty_iff(2)
thf(fact_2625_less__eq__option__def,axiom,
    ( ord_le353528952715127954et_int
    = ( ^ [X4: option_set_int,Y4: option_set_int] :
          ( case_o223999843215110191et_int @ $true
          @ ^ [Z6: set_int] : ( case_o223999843215110191et_int @ $false @ ( ord_less_eq_set_int @ Z6 ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_2626_less__eq__option__def,axiom,
    ( ord_le2406147912482264968on_rat
    = ( ^ [X4: option_rat,Y4: option_rat] :
          ( case_option_o_rat @ $true
          @ ^ [Z6: rat] : ( case_option_o_rat @ $false @ ( ord_less_eq_rat @ Z6 ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_2627_less__eq__option__def,axiom,
    ( ord_le6622620407824499402on_num
    = ( ^ [X4: option_num,Y4: option_num] :
          ( case_option_o_num @ $true
          @ ^ [Z6: num] : ( case_option_o_num @ $false @ ( ord_less_eq_num @ Z6 ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_2628_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z6: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z6 ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_2629_less__eq__option__def,axiom,
    ( ord_le1736525451366464988on_int
    = ( ^ [X4: option_int,Y4: option_int] :
          ( case_option_o_int @ $true
          @ ^ [Z6: int] : ( case_option_o_int @ $false @ ( ord_less_eq_int @ Z6 ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_2630_less__option__def,axiom,
    ( ord_less_option_assn
    = ( ^ [X4: option_assn] :
          ( case_option_o_assn @ $false
          @ ^ [Y4: assn] :
              ( case_option_o_assn @ $true
              @ ^ [Z6: assn] : ( ord_less_assn @ Z6 @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_2631_less__option__def,axiom,
    ( ord_less_option_rat
    = ( ^ [X4: option_rat] :
          ( case_option_o_rat @ $false
          @ ^ [Y4: rat] :
              ( case_option_o_rat @ $true
              @ ^ [Z6: rat] : ( ord_less_rat @ Z6 @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_2632_less__option__def,axiom,
    ( ord_less_option_num
    = ( ^ [X4: option_num] :
          ( case_option_o_num @ $false
          @ ^ [Y4: num] :
              ( case_option_o_num @ $true
              @ ^ [Z6: num] : ( ord_less_num @ Z6 @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_2633_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X4: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y4: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z6: nat] : ( ord_less_nat @ Z6 @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_2634_less__option__def,axiom,
    ( ord_less_option_int
    = ( ^ [X4: option_int] :
          ( case_option_o_int @ $false
          @ ^ [Y4: int] :
              ( case_option_o_int @ $true
              @ ^ [Z6: int] : ( ord_less_int @ Z6 @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_2635_Heap__cases,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
      ( ! [X3: product_unit,H5: produc6653097349344004940it_nat] :
          ( ( heap_T875086893843062177t_unit @ F @ H )
         != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X3 @ H5 ) ) )
     => ( ( heap_T875086893843062177t_unit @ F @ H )
        = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
thf(fact_2636_Heap__cases,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ! [X3: a,H5: produc6653097349344004940it_nat] :
          ( ( heap_Time_execute_a @ F @ H )
         != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X3 @ H5 ) ) )
     => ( ( heap_Time_execute_a @ F @ H )
        = none_P2651198173097904984it_nat ) ) ).

% Heap_cases
thf(fact_2637_timeFrame_Ocases,axiom,
    ! [X: produc4453839368661128058it_nat] :
      ( ! [N5: nat,R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] :
          ( X
         != ( produc4111269673004989362it_nat @ N5 @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) ) )
     => ~ ! [N5: nat] :
            ( X
           != ( produc4111269673004989362it_nat @ N5 @ none_P2651198173097904984it_nat ) ) ) ).

% timeFrame.cases
thf(fact_2638_timeFrame_Ocases,axiom,
    ! [X: produc3911288613690379145it_nat] :
      ( ! [N5: nat,R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] :
          ( X
         != ( produc638857205735767105it_nat @ N5 @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) ) )
     => ~ ! [N5: nat] :
            ( X
           != ( produc638857205735767105it_nat @ N5 @ none_P9117596204409417319it_nat ) ) ) ).

% timeFrame.cases
thf(fact_2639_set__to__map__empty__iff_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( set_to_map_nat_nat @ S )
        = ( ^ [X4: nat] : none_nat ) )
      = ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(1)
thf(fact_2640_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( ( As4 = bot_bot_set_nat )
              & X ) ) ) ).

% pure_assn_raw.elims(3)
thf(fact_2641_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ~ ( ( As4 = bot_bot_set_nat )
                & X ) ) ) ).

% pure_assn_raw.elims(2)
thf(fact_2642_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( Y
              = ( ~ ( ( As4 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.elims(1)
thf(fact_2643_pure__assn__raw_Osimps,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( pure_a825153325127701367it_nat @ B @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ( As = bot_bot_set_nat )
        & B ) ) ).

% pure_assn_raw.simps
thf(fact_2644_pure__assn__raw_Opelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( Y
                  = ( ( As4 = bot_bot_set_nat )
                    & X ) )
               => ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ) ) ) ).

% pure_assn_raw.pelims(1)
thf(fact_2645_pure__assn__raw_Opelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) )
               => ~ ( ( As4 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
thf(fact_2646_pure__assn__raw_Opelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) )
               => ( ( As4 = bot_bot_set_nat )
                  & X ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
thf(fact_2647_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H3: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_Time_Heap_a] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( heap_Time_execute_a @ ( heap_T757603679106148408unit_a @ F @ G ) @ H )
        = ( heap_T7616092557645711335rame_a @ N2 @ ( heap_Time_execute_a @ ( G @ X ) @ H3 ) ) ) ) ).

% execute_bind(1)
thf(fact_2648_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H3: heap_e7401611519738050253t_unit,N2: nat,G: a > heap_Time_Heap_a] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( heap_Time_execute_a @ ( heap_Time_bind_a_a @ F @ G ) @ H )
        = ( heap_T7616092557645711335rame_a @ N2 @ ( heap_Time_execute_a @ ( G @ X ) @ H3 ) ) ) ) ).

% execute_bind(1)
thf(fact_2649_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H3: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T5738788834812785303t_unit,Y: product_unit,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H3 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_2650_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H3: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_Time_Heap_a,Y: a,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_a @ ( G @ X ) @ H3 )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_a @ ( heap_T757603679106148408unit_a @ F @ G ) @ H )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_2651_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H3: heap_e7401611519738050253t_unit,N2: nat,G: a > heap_T5738788834812785303t_unit,Y: product_unit,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H3 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_T875086893843062177t_unit @ ( heap_T8685611227969916822t_unit @ F @ G ) @ H )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_2652_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H3: heap_e7401611519738050253t_unit,N2: nat,G: a > heap_Time_Heap_a,Y: a,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_a @ ( G @ X ) @ H3 )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_a @ ( heap_Time_bind_a_a @ F @ G ) @ H )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_2653_set__to__map__def,axiom,
    ( set_to_map_nat_nat
    = ( ^ [S5: set_Pr1261947904930325089at_nat,K3: nat] :
          ( eps_Opt_nat
          @ ^ [V2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2654_set__to__map__def,axiom,
    ( set_to6040779677306527128et_nat
    = ( ^ [S5: set_Pr3286484037609594932et_nat,K3: produc3658429121746597890et_nat > $o] :
          ( eps_Op2013419657081471078et_nat
          @ ^ [V2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2655_set__to__map__def,axiom,
    ( set_to2047380710992656148et_nat
    = ( ^ [S5: set_Pr8536935166611901872et_nat,K3: produc3658429121746597890et_nat > $o] :
          ( eps_Op6423059015816587746et_nat
          @ ^ [V2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2656_set__to__map__def,axiom,
    ( set_to6585427216324510421nt_int
    = ( ^ [S5: set_Pr9222295170931077689nt_int,K3: produc8551481072490612790e_term > option6357759511663192854e_term] :
          ( eps_Op2446201859369042517nt_int
          @ ^ [V2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2657_set__to__map__def,axiom,
    ( set_to1653371902132773855nt_int
    = ( ^ [S5: set_Pr1872883991513573699nt_int,K3: int > option6357759511663192854e_term] :
          ( eps_Op2446201859369042517nt_int
          @ ^ [V2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2658_set__to__map__def,axiom,
    ( set_to2008915276733485075it_nat
    = ( ^ [S5: set_Pr7098220151150636591it_nat,K3: a] :
          ( eps_Op8301357815426737072it_nat
          @ ^ [V2: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ K3 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_2659_execute__guard_I2_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( P @ H )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H )
        = ( some_P1914260805536162275it_nat @ ( F @ H ) ) ) ) ).

% execute_guard(2)
thf(fact_2660_execute__guard_I2_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
      ( ( P @ H )
     => ( ( heap_Time_execute_a @ ( heap_Time_guard_a @ P @ F ) @ H )
        = ( some_P7913643980934408916it_nat @ ( F @ H ) ) ) ) ).

% execute_guard(2)
thf(fact_2661_mlex__leq,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_2662_curryI,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( produc6216949301066131538_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_2663_curryI,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( produc5101573711933517782_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_2664_curryI,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( produc730925184835016917_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_2665_curryI,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( produc9098658269643458507_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_2666_curryI,axiom,
    ! [F: produc3260487557148687353it_nat > $o,A: a,B: produc6653097349344004940it_nat] :
      ( ( F @ ( produc9178034014595674355it_nat @ A @ B ) )
     => ( produc3709033845735585303_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_2667_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_2668_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_2669_some__opt__sym__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] :
      ( ( eps_Op1741419982095519837it_nat
        @ ( ^ [Y5: produc3260487557148687353it_nat,Z3: produc3260487557148687353it_nat] : Y5 = Z3
          @ X ) )
      = ( some_P7913643980934408916it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_2670_some__opt__sym__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ( ^ [Y5: num,Z3: num] : Y5 = Z3
          @ X ) )
      = ( some_num @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_2671_some__opt__sym__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat
        @ ( ^ [Y5: produc8664842809031399944it_nat,Z3: produc8664842809031399944it_nat] : Y5 = Z3
          @ X ) )
      = ( some_P1914260805536162275it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_2672_some__opt__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] :
      ( ( eps_Op1741419982095519837it_nat
        @ ^ [Y4: produc3260487557148687353it_nat] : Y4 = X )
      = ( some_P7913643980934408916it_nat @ X ) ) ).

% some_opt_eq_trivial
thf(fact_2673_some__opt__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ^ [Y4: num] : Y4 = X )
      = ( some_num @ X ) ) ).

% some_opt_eq_trivial
thf(fact_2674_some__opt__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat
        @ ^ [Y4: produc8664842809031399944it_nat] : Y4 = X )
      = ( some_P1914260805536162275it_nat @ X ) ) ).

% some_opt_eq_trivial
thf(fact_2675_some__opt__false__trivial,axiom,
    ( ( eps_Opt_num
      @ ^ [Uu2: num] : $false )
    = none_num ) ).

% some_opt_false_trivial
thf(fact_2676_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_2677_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_2678_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_2679_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_2680_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_2681_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_2682_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_2683_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_2684_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_2685_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_2686_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_2687_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_2688_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_2689_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N5: nat] :
          ( L
          = ( plus_plus_nat @ K @ N5 ) ) ) ).

% le_Suc_ex
thf(fact_2690_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_2691_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_2692_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_2693_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_2694_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_2695_curryE,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_2696_curryE,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_2697_curryE,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_2698_curryE,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_2699_curryE,axiom,
    ! [F: produc3260487557148687353it_nat > $o,A: a,B: produc6653097349344004940it_nat] :
      ( ( produc3709033845735585303_nat_o @ F @ A @ B )
     => ( F @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ).

% curryE
thf(fact_2700_curryD,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_2701_curryD,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_2702_curryD,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_2703_curryD,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_2704_curryD,axiom,
    ! [F: produc3260487557148687353it_nat > $o,A: a,B: produc6653097349344004940it_nat] :
      ( ( produc3709033845735585303_nat_o @ F @ A @ B )
     => ( F @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ).

% curryD
thf(fact_2705_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R3: a,H: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( heap_T7616092557645711335rame_a @ N2 @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R3 @ ( produc584006145561248582it_nat @ H @ N6 ) ) ) )
      = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R3 @ ( produc584006145561248582it_nat @ H @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_2706_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R3: product_unit,H: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( heap_T3616969660504097270t_unit @ N2 @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R3 @ ( produc584006145561248582it_nat @ H @ N6 ) ) ) )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R3 @ ( produc584006145561248582it_nat @ H @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_2707_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( ( heap_T7616092557645711335rame_a @ X @ Xa )
        = Y )
     => ( ! [R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa
              = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) )
           => ( Y
             != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ ( plus_plus_nat @ X @ N4 ) ) ) ) ) )
       => ~ ( ( Xa = none_P2651198173097904984it_nat )
           => ( Y != none_P2651198173097904984it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_2708_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( ( heap_T3616969660504097270t_unit @ X @ Xa )
        = Y )
     => ( ! [R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa
              = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) )
           => ( Y
             != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ ( plus_plus_nat @ X @ N4 ) ) ) ) ) )
       => ~ ( ( Xa = none_P9117596204409417319it_nat )
           => ( Y != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_2709_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N5: nat] :
          ( ( ord_less_nat @ M5 @ N5 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N5 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2710_Eps__Opt__eq__Some,axiom,
    ! [P: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ! [X9: produc3260487557148687353it_nat] :
          ( ( P @ X )
         => ( ( P @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Op1741419982095519837it_nat @ P )
          = ( some_P7913643980934408916it_nat @ X ) )
        = ( P @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_2711_Eps__Opt__eq__Some,axiom,
    ! [P: num > $o,X: num] :
      ( ! [X9: num] :
          ( ( P @ X )
         => ( ( P @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Opt_num @ P )
          = ( some_num @ X ) )
        = ( P @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_2712_Eps__Opt__eq__Some,axiom,
    ! [P: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ! [X9: produc8664842809031399944it_nat] :
          ( ( P @ X )
         => ( ( P @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Op3393321821070424684it_nat @ P )
          = ( some_P1914260805536162275it_nat @ X ) )
        = ( P @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_2713_Eps__Opt__eq__Some__implies,axiom,
    ! [P: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ( ( eps_Op1741419982095519837it_nat @ P )
        = ( some_P7913643980934408916it_nat @ X ) )
     => ( P @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_2714_Eps__Opt__eq__Some__implies,axiom,
    ! [P: num > $o,X: num] :
      ( ( ( eps_Opt_num @ P )
        = ( some_num @ X ) )
     => ( P @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_2715_Eps__Opt__eq__Some__implies,axiom,
    ! [P: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ( ( eps_Op3393321821070424684it_nat @ P )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( P @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_2716_execute__guard_I1_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
      ( ~ ( P @ H )
     => ( ( heap_Time_execute_a @ ( heap_Time_guard_a @ P @ F ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_guard(1)
thf(fact_2717_mlex__less,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R ) ) ) ).

% mlex_less
thf(fact_2718_mlex__iff,axiom,
    ! [X: nat,Y: nat,F: nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_2719_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( ( heap_T7616092557645711335rame_a @ X @ Xa )
        = Y )
     => ( ( accp_P6263092265436569219it_nat @ heap_T8132184524487034138_rel_a @ ( produc4111269673004989362it_nat @ X @ Xa ) )
       => ( ! [R2: a,H4: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa
                = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ ( plus_plus_nat @ X @ N4 ) ) ) ) )
               => ~ ( accp_P6263092265436569219it_nat @ heap_T8132184524487034138_rel_a @ ( produc4111269673004989362it_nat @ X @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) ) ) ) )
         => ~ ( ( Xa = none_P2651198173097904984it_nat )
             => ( ( Y = none_P2651198173097904984it_nat )
               => ~ ( accp_P6263092265436569219it_nat @ heap_T8132184524487034138_rel_a @ ( produc4111269673004989362it_nat @ X @ none_P2651198173097904984it_nat ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_2720_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( ( heap_T3616969660504097270t_unit @ X @ Xa )
        = Y )
     => ( ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ Xa ) )
       => ( ! [R2: product_unit,H4: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa
                = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ ( plus_plus_nat @ X @ N4 ) ) ) ) )
               => ~ ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H4 @ N4 ) ) ) ) ) ) )
         => ~ ( ( Xa = none_P9117596204409417319it_nat )
             => ( ( Y = none_P9117596204409417319it_nat )
               => ~ ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ none_P9117596204409417319it_nat ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_2721_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2722_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2723_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_2724_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2725_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2726_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_2727_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2728_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2729_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2730_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2731_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2732_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2733_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2734_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2735_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2736_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2737_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2738_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2739_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2740_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2741_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2742_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2743_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2744_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2745_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2746_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2747_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2748_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2749_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_2750_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_2751_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2752_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2753_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2754_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_2755_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2756_mlex__bound,axiom,
    ! [A: nat,A3: nat,B: nat,N7: nat] :
      ( ( ord_less_nat @ A @ A3 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( times_times_nat @ A3 @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_2757_mlex__fst__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,N7: nat,B2: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ( ord_less_nat @ B2 @ N7 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_2758_mlex__snd__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( ( A = A2 )
     => ( ( ord_less_nat @ B @ B2 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_2759_mlex__leI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( ( ord_less_eq_nat @ A @ A2 )
     => ( ( ord_less_eq_nat @ B @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) ) ) ) ).

% mlex_leI
thf(fact_2760_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2761_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2762_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2763_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2764_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2765_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_3573771949741848930nteger
    = ( ^ [A5: code_integer,B5: code_integer] : ( times_3573771949741848930nteger @ B5 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2766_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A5: assn,B5: assn] : ( times_times_assn @ B5 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2767_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ B5 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2768_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2769_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2770_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( times_3573771949741848930nteger @ A @ C ) )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2771_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( times_times_assn @ B @ ( times_times_assn @ A @ C ) )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2772_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2773_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2774_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2775_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2776_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2777_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2778_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2779_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2780_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2781_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
        ? [C4: nat] :
          ( B5
          = ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_2782_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2783_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2784_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2785_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_2786_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2787_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2788_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2789_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2790_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2791_add__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2792_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2793_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2794_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2795_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2796_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2797_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2798_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2799_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2800_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2801_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2802_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2803_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_2804_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2805_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2806_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_2807_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2808_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2809_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2810_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2811_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2812_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_2813_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2814_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2815_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2816_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2817_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2818_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2819_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2820_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2821_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2822_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2823_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2824_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2825_crossproduct__eq,axiom,
    ! [W2: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W2 @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W2 @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2826_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2827_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2828_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2829_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2830_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2832_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2833_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2834_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2835_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2836_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2837_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2838_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2839_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% distrib_left
thf(fact_2840_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2841_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2842_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2843_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% distrib_right
thf(fact_2844_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2845_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2846_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_2847_combine__common__factor,axiom,
    ! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2848_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2849_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2850_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2851_execute__raise,axiom,
    ! [S2: list_char] :
      ( ( heap_Time_execute_a @ ( heap_Time_raise_a @ S2 ) )
      = ( ^ [Uu2: heap_e7401611519738050253t_unit] : none_P2651198173097904984it_nat ) ) ).

% execute_raise
thf(fact_2852_guard__def,axiom,
    ( heap_Time_guard_a
    = ( ^ [P4: heap_e7401611519738050253t_unit > $o,F3: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
          ( heap_Time_Heap_a2
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( if_opt6883606601682554499it_nat @ ( P4 @ H6 ) @ ( some_P7913643980934408916it_nat @ ( F3 @ H6 ) ) @ none_P2651198173097904984it_nat ) ) ) ) ).

% guard_def
thf(fact_2853_guard__def,axiom,
    ( heap_T8440541562793052209t_unit
    = ( ^ [P4: heap_e7401611519738050253t_unit > $o,F3: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( if_opt1729522071442692626it_nat @ ( P4 @ H6 ) @ ( some_P1914260805536162275it_nat @ ( F3 @ H6 ) ) @ none_P9117596204409417319it_nat ) ) ) ) ).

% guard_def
thf(fact_2854_execute__assert_I2_J,axiom,
    ! [P: a > $o,X: a,H: heap_e7401611519738050253t_unit] :
      ( ~ ( P @ X )
     => ( ( heap_Time_execute_a @ ( heap_Time_assert_a @ P @ X ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_assert(2)
thf(fact_2855_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H6 )
           => B )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_2856_Heap__execute,axiom,
    ! [F: heap_Time_Heap_a] :
      ( ( heap_Time_Heap_a2 @ ( heap_Time_execute_a @ F ) )
      = F ) ).

% Heap_execute
thf(fact_2857_Heap__execute,axiom,
    ! [F: heap_T5738788834812785303t_unit] :
      ( ( heap_T6183433275982383450t_unit @ ( heap_T875086893843062177t_unit @ F ) )
      = F ) ).

% Heap_execute
thf(fact_2858_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( B
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_2859_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H6 ) ) ) ).

% ent_false_iff
thf(fact_2860_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_2861_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_2862_ent__iffI,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% ent_iffI
thf(fact_2863_raise__def,axiom,
    ( heap_T2927564422264180874t_unit
    = ( ^ [S6: list_char] :
          ( heap_T6183433275982383450t_unit
          @ ^ [Uu2: heap_e7401611519738050253t_unit] : none_P9117596204409417319it_nat ) ) ) ).

% raise_def
thf(fact_2864_entails__def,axiom,
    ( entails
    = ( ^ [P4: assn,Q3: assn] :
        ! [H6: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P4 @ H6 )
         => ( rep_assn @ Q3 @ H6 ) ) ) ) ).

% entails_def
thf(fact_2865_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H4 )
         => ( rep_assn @ Q @ H4 ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_2866_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H )
       => ( rep_assn @ Q @ H ) ) ) ).

% entailsD
thf(fact_2867_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H ) ) ) ).

% ent_fwd
thf(fact_2868_ent__star__mono,axiom,
    ! [P: assn,P6: assn,Q: assn,Q4: assn] :
      ( ( entails @ P @ P6 )
     => ( ( entails @ Q @ Q4 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P6 @ Q4 ) ) ) ) ).

% ent_star_mono
thf(fact_2869_ent__disjE,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( ( entails @ B3 @ C3 )
       => ( entails @ ( sup_sup_assn @ A3 @ B3 ) @ C3 ) ) ) ).

% ent_disjE
thf(fact_2870_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ P @ R ) ) ).

% ent_disjI1
thf(fact_2871_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ Q @ R ) ) ).

% ent_disjI2
thf(fact_2872_ent__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% ent_disjI1'
thf(fact_2873_ent__disjI2_H,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% ent_disjI2'
thf(fact_2874_ent__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : ( entails @ A3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI1_direct
thf(fact_2875_ent__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : ( entails @ B3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI2_direct
thf(fact_2876_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option3562590408128118217it_nat] :
      ( ( heap_Time_execute_a @ ( heap_Time_Heap_a2 @ F ) )
      = F ) ).

% execute.simps
thf(fact_2877_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T6183433275982383450t_unit @ F ) )
      = F ) ).

% execute.simps
thf(fact_2878_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_2879_ent__conjI,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ A3 @ C3 )
       => ( entails @ A3 @ ( inf_inf_assn @ B3 @ C3 ) ) ) ) ).

% ent_conjI
thf(fact_2880_ent__conjE1,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C3 ) ) ).

% ent_conjE1
thf(fact_2881_ent__conjE2,axiom,
    ! [B3: assn,C3: assn,A3: assn] :
      ( ( entails @ B3 @ C3 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C3 ) ) ).

% ent_conjE2
thf(fact_2882_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_2883_ent__mp,axiom,
    ! [P: assn,Q: assn] : ( entails @ ( times_times_assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).

% ent_mp
thf(fact_2884_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_2885_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_2886_execute__assert_I1_J,axiom,
    ! [P: product_unit > $o,X: product_unit,H: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_2887_execute__assert_I1_J,axiom,
    ! [P: a > $o,X: a,H: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( ( heap_Time_execute_a @ ( heap_Time_assert_a @ P @ X ) @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_2888_Heap__lub__empty,axiom,
    ( ( heap_T4436805183663264329t_unit @ bot_bo8105976889876132193t_unit )
    = ( heap_T6183433275982383450t_unit
      @ ^ [X4: heap_e7401611519738050253t_unit] : none_P9117596204409417319it_nat ) ) ).

% Heap_lub_empty
thf(fact_2889_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( ( entails @ P @ ( pure_assn @ B ) )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H6 )
           => B )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_2890_pairself_Opelims,axiom,
    ! [X: int > option6357759511663192854e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
      ( ( ( pairse3534876335208825186e_term @ X @ Xa )
        = Y )
     => ( ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ Xa ) )
       => ~ ! [A4: int,B4: int] :
              ( ( Xa
                = ( product_Pair_int_int @ A4 @ B4 ) )
             => ( ( Y
                  = ( produc5950683997804057413e_term @ ( X @ A4 ) @ ( X @ B4 ) ) )
               => ~ ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ ( product_Pair_int_int @ A4 @ B4 ) ) ) ) ) ) ) ).

% pairself.pelims
thf(fact_2891_one__assn__raw_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( As = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_2892_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( Y
              = ( As4 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_2893_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( As4 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_2894_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
           => ( As4 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_2895_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2896_combine__options__def,axiom,
    ( combin5819755802923621690it_nat
    = ( ^ [F3: produc3260487557148687353it_nat > produc3260487557148687353it_nat > produc3260487557148687353it_nat,X4: option3562590408128118217it_nat,Y4: option3562590408128118217it_nat] :
          ( case_o6061097939050036047it_nat @ Y4
          @ ^ [Z6: produc3260487557148687353it_nat] :
              ( case_o6061097939050036047it_nat @ ( some_P7913643980934408916it_nat @ Z6 )
              @ ^ [Aa: produc3260487557148687353it_nat] : ( some_P7913643980934408916it_nat @ ( F3 @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_2897_combine__options__def,axiom,
    ( combin4318129983670048329it_nat
    = ( ^ [F3: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,X4: option8956607266484857688it_nat,Y4: option8956607266484857688it_nat] :
          ( case_o2963978774867076333it_nat @ Y4
          @ ^ [Z6: produc8664842809031399944it_nat] :
              ( case_o2963978774867076333it_nat @ ( some_P1914260805536162275it_nat @ Z6 )
              @ ^ [Aa: produc8664842809031399944it_nat] : ( some_P1914260805536162275it_nat @ ( F3 @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_2898_combine__options__def,axiom,
    ( combine_options_num
    = ( ^ [F3: num > num > num,X4: option_num,Y4: option_num] :
          ( case_o6005452278849405969um_num @ Y4
          @ ^ [Z6: num] :
              ( case_o6005452278849405969um_num @ ( some_num @ Z6 )
              @ ^ [Aa: num] : ( some_num @ ( F3 @ Z6 @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_2899_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_2900_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_2901_mult__zero__left,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_2902_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_2903_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2904_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2905_mult__zero__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_2906_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_2907_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2908_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2909_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_2910_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_2911_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_2912_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_2913_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2914_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2915_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2916_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2917_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2918_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2919_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2920_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2921_mult_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.right_neutral
thf(fact_2922_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_2923_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_2924_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2925_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2926_mult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% mult_1
thf(fact_2927_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_2928_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_2929_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2930_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2931_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_2932_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_2933_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_2934_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_2935_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_2936_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_2937_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2938_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2939_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2940_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2941_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2942_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2943_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2944_combine__options__simps_I3_J,axiom,
    ! [F: produc3260487557148687353it_nat > produc3260487557148687353it_nat > produc3260487557148687353it_nat,A: produc3260487557148687353it_nat,B: produc3260487557148687353it_nat] :
      ( ( combin5819755802923621690it_nat @ F @ ( some_P7913643980934408916it_nat @ A ) @ ( some_P7913643980934408916it_nat @ B ) )
      = ( some_P7913643980934408916it_nat @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_2945_combine__options__simps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( combine_options_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_2946_combine__options__simps_I3_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,A: produc8664842809031399944it_nat,B: produc8664842809031399944it_nat] :
      ( ( combin4318129983670048329it_nat @ F @ ( some_P1914260805536162275it_nat @ A ) @ ( some_P1914260805536162275it_nat @ B ) )
      = ( some_P1914260805536162275it_nat @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_2947_combine__options__simps_I2_J,axiom,
    ! [F: num > num > num,X: option_num] :
      ( ( combine_options_num @ F @ X @ none_num )
      = X ) ).

% combine_options_simps(2)
thf(fact_2948_combine__options__simps_I1_J,axiom,
    ! [F: num > num > num,Y: option_num] :
      ( ( combine_options_num @ F @ none_num @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_2949_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_2950_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_2951_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_2952_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2953_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2954_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2955_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2956_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2957_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2958_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel2
thf(fact_2959_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2960_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2961_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_2962_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel1
thf(fact_2963_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2964_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2965_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_2966_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_2967_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2968_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2969_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2970_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_2971_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2972_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2973_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2974_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2975_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2976_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2977_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2978_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2979_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2980_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel2
thf(fact_2981_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2982_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2983_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_2984_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel1
thf(fact_2985_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2986_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2987_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_2988_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_2989_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_2990_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_2991_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_2992_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_2993_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_2994_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_2995_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_2996_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_2997_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_2998_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2999_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_3000_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_3001_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_3002_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_3003_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_3004_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_3005_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_3006_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_3007_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_3008_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_3009_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_3010_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_3011_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_3012_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_3022_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_3023_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_3024_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_3025_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_3026_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_3027_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_3028_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_3029_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_3030_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_3031_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_3032_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_3033_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_3034_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_3035_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3036_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_3037_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3038_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3039_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3040_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3041_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_3042_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_3043_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3044_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_3052_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3053_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3054_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3055_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_3056_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3057_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3058_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3059_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3060_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_3061_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.comm_neutral
thf(fact_3062_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_3063_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_3064_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_3065_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_3066_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_3067_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_3068_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_3069_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_3070_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
       != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_3071_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_3072_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_3073_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_3074_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_3075_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_3076_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_3077_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_3078_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A @ B )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_3079_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_3080_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_3081_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_3082_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C @ A )
          = ( times_3573771949741848930nteger @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3083_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3084_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3085_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3086_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A @ C )
          = ( times_3573771949741848930nteger @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3087_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3088_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3089_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3090_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ~ ( P @ N5 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N5 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_3091_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_3092_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_3093_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_3094_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_3095_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_3096_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_3097_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_3098_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_3099_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_3100_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_3101_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_3102_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_3103_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_3104_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_3105_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_3106_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_3107_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3108_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3109_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3110_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3111_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3112_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3113_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3114_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3115_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3116_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3117_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3118_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3119_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3120_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3121_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3122_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3123_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3124_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3125_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3126_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3127_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3128_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3129_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3130_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3131_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3132_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3133_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3134_lambda__one,axiom,
    ( ( ^ [X4: code_integer] : X4 )
    = ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).

% lambda_one
thf(fact_3135_lambda__one,axiom,
    ( ( ^ [X4: assn] : X4 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_3136_lambda__one,axiom,
    ( ( ^ [X4: rat] : X4 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_3137_lambda__one,axiom,
    ( ( ^ [X4: nat] : X4 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_3138_lambda__one,axiom,
    ( ( ^ [X4: int] : X4 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_3139_lambda__zero,axiom,
    ( ( ^ [H6: code_integer] : zero_z3403309356797280102nteger )
    = ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger ) ) ).

% lambda_zero
thf(fact_3140_lambda__zero,axiom,
    ( ( ^ [H6: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_3141_lambda__zero,axiom,
    ( ( ^ [H6: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_3142_lambda__zero,axiom,
    ( ( ^ [H6: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_3143_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3144_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3145_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3146_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_3147_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3148_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3149_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3150_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_3151_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_3152_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_3153_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_3154_less__1__mult,axiom,
    ! [M: code_integer,N2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N2 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3155_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3156_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3157_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_3167_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3168_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3169_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3170_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3171_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3172_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3173_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3174_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3175_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3176_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3177_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3178_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3179_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3180_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3181_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3182_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3183_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3184_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3185_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3186_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3187_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3188_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3189_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3190_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3191_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3192_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3193_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3194_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3195_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3196_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3197_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3198_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3199_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3200_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3201_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3202_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3203_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3204_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3205_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3206_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3207_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3208_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3209_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3210_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3211_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3212_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3213_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_3214_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_3215_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_3216_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_3217_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3218_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3219_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3220_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3221_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3222_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3223_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3224_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3225_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3226_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3227_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3228_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3229_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3230_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3231_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3232_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3233_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3234_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3235_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3236_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3237_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3238_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3239_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3240_zero__le__square,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).

% zero_le_square
thf(fact_3241_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_3242_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_3243_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3244_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3245_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3246_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3247_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3248_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3249_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3250_mult__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3251_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3252_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3253_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3254_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3255_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3256_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3257_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3258_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3259_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3260_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_3261_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3262_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3263_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3264_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_3265_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_3266_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_3267_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3268_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3269_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3270_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3271_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3272_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3273_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3274_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3275_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3276_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3277_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3278_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3279_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3280_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3281_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3282_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3283_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3284_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3285_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3286_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3287_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3288_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3289_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3290_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3291_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3292_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3293_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3294_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3295_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3296_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3297_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3298_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3299_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3300_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3301_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3302_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3303_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3304_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3305_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_3306_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_3307_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_3308_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_3309_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_3310_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_3311_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_3312_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3313_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3314_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3315_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3316_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_3317_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_3318_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_3319_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_3320_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_3321_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_3322_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_3323_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_3324_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_3325_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_3326_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_3327_not__square__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_3328_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_3329_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_3330_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_3331_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_3332_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_3333_add__scale__eq__noteq,axiom,
    ! [R3: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R3 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R3 @ C ) )
         != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3334_add__scale__eq__noteq,axiom,
    ! [R3: rat,A: rat,B: rat,C: rat,D2: rat] :
      ( ( R3 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R3 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3335_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C: nat,D2: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3336_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C: int,D2: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3337_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_3338_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3339_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_3340_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_3341_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_3342_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3343_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3344_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3345_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3346_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3347_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3348_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3349_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3350_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3351_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3352_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3353_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3354_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_3355_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3356_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3357_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3358_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3359_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3360_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3361_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3362_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_3363_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3364_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3365_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3366_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3367_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3368_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3369_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3370_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3371_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3372_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3373_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3374_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3375_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3376_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3377_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3378_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3379_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3380_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3381_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3382_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3383_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3384_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3385_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3386_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3387_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3388_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3389_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3390_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3391_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3392_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3393_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3394_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3395_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3396_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3397_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3398_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3399_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3400_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3401_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3402_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3403_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3404_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3405_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3406_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3407_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3408_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3409_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3410_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3411_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3412_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3413_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3414_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3415_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3416_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_3417_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_3418_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_3419_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_3420_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_3421_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_3422_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_3423_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_3424_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_3425_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3426_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3427_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A5 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_3428_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3429_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_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_3442_linordered__field__no__lb,axiom,
    ! [X7: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X7 ) ).

% linordered_field_no_lb
thf(fact_3443_linordered__field__no__ub,axiom,
    ! [X7: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X7 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_3444_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_3445_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_3446_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_3447_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_3448_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_3449_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_3450_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_3451_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_3452_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_3453_divides__aux__eq,axiom,
    ! [Q6: code_integer,R3: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q6 @ R3 ) )
      = ( R3 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_3454_divides__aux__eq,axiom,
    ! [Q6: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q6 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3455_divides__aux__eq,axiom,
    ! [Q6: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q6 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3456_tap__def,axiom,
    ( heap_Time_tap_a
    = ( ^ [F3: heap_e7401611519738050253t_unit > a] :
          ( heap_Time_Heap_a2
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ ( F3 @ H6 ) @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_3457_tap__def,axiom,
    ( heap_T560649228465745139t_unit
    = ( ^ [F3: heap_e7401611519738050253t_unit > product_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F3 @ H6 ) @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_3458_execute__tap,axiom,
    ! [F: heap_e7401611519738050253t_unit > product_unit,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T560649228465745139t_unit @ F ) @ H )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_tap
thf(fact_3459_execute__tap,axiom,
    ! [F: heap_e7401611519738050253t_unit > a,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_a @ ( heap_Time_tap_a @ F ) @ H )
      = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ ( F @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_tap
thf(fact_3460_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
               => ( As4 = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_3461_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
               => ( As4 != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_3462_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As4 ) )
             => ( ( Y
                  = ( As4 = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)
thf(fact_3463_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_3464_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_3465_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_3466_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_3467_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_3468_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_3469_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_3470_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_3471_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M6: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M6 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X7: nat] :
                    ( ( P @ X7 )
                   => ( ord_less_eq_nat @ X7 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3472_le__numeral__extra_I3_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% le_numeral_extra(3)
thf(fact_3473_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_3474_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_3475_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_3476_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_numeral_extra(3)
thf(fact_3477_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_3478_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_3479_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_3480_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_3481_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_3482_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_3483_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_3484_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3485_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3486_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3487_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3488_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A5: nat,B5: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B5 @ A5 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B5 @ ( F3 @ A5 @ Acc ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_3489_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_3490_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_3491_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_3492_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_3493_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_3494_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3495_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3496_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3497_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3498_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_3499_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_3500_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_3501_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_3502_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3503_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_3504_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_3505_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_3506_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3507_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3508_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3509_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3510_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3511_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3512_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3513_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3514_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3515_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3516_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3517_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3518_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3519_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3520_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3521_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M )
          = ( power_8256067586552552935nteger @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_3522_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_3523_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_3524_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_3525_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_3526_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_3527_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_3528_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_3529_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_3530_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_3531_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_3532_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_3533_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_3534_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_3535_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3536_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3537_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3538_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3539_power__eq__0__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ A @ N2 )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_3540_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( power_power_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_3541_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_3542_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_3543_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_3544_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3545_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3546_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3547_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3548_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
            = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3549_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3550_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3551_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_3552_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_3553_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N2: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_3554_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_3555_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_3556_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_3557_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_3558_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N2: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A @ B ) @ N2 )
      = ( times_times_assn @ ( power_power_assn @ A @ N2 ) @ ( power_power_assn @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_3559_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_3560_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_3561_power__mult__distrib,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_3562_power__commutes,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ A )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_3563_power__commutes,axiom,
    ! [A: assn,N2: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A @ N2 ) @ A )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_3564_power__commutes,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_3565_power__commutes,axiom,
    ! [A: nat,N2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_3566_power__commutes,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_3567_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3568_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_3569_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3570_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_3571_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_3572_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_3573_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_3574_zero__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_3575_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_3576_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_3577_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_3578_zero__less__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_3579_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_3580_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_3581_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_3582_one__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_3583_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_3584_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_3585_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_3586_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_3587_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_3588_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_3589_divide__nonneg__nonneg,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 @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_3590_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_3591_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_3592_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_3593_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3594_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_3595_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_3596_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_3597_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_3598_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_3599_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_3600_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_3601_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_3602_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_3603_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_3604_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N2: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ ( power_power_assn @ Y @ N2 ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_3605_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_3606_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3607_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3608_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W2 @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3609_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_3610_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_3611_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3612_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_3613_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3614_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3615_power__add,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_add
thf(fact_3616_power__add,axiom,
    ! [A: assn,M: nat,N2: nat] :
      ( ( power_power_assn @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_assn @ ( power_power_assn @ A @ M ) @ ( power_power_assn @ A @ N2 ) ) ) ).

% power_add
thf(fact_3617_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_3618_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_3619_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_3620_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_3621_power__less__imp__less__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3622_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3623_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3624_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3625_power__le__one,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_3626_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3627_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3628_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3629_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_3630_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_3631_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_3632_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3633_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_3634_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3635_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3636_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_3637_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3638_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3639_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3640_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3641_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3642_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3643_power__gt1__lemma,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3644_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3645_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3646_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3647_power__less__power__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3648_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3649_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3650_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3651_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3652_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3653_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3654_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3655_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3656_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3657_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3658_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3659_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3660_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3661_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3662_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3663_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3664_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_3665_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_3666_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3667_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3668_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3669_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3670_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3671_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3672_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3673_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3674_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3675_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3676_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3677_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3678_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3679_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3680_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3681_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3682_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3683_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3684_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3685_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_3686_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_rat @ zero_zero_rat @ N2 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_3687_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_3688_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_3689_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_3690_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_3691_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3692_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_3693_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_3694_power__Suc__less,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_3695_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_3696_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_3697_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_3698_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3699_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3700_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3701_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3702_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3703_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3704_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3705_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3706_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3707_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3708_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_3709_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_3710_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N7 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3711_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3712_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3713_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3714_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N7 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_3715_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_3716_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_3717_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_3718_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ( power_8256067586552552935nteger @ A @ N2 )
              = ( power_8256067586552552935nteger @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3719_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N2 )
              = ( power_power_rat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3720_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3721_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3722_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ N2 )
        = ( power_8256067586552552935nteger @ B @ N2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3723_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N2 )
        = ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3724_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3725_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3726_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3727_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3728_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3729_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3730_self__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_3731_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_3732_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_3733_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_3734_one__less__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_3735_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_3736_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_3737_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_3738_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3739_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3740_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3741_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3742_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3743_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3744_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3745_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3746_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3747_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3748_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3749_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3750_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3751_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3752_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3753_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3754_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_3755_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_3756_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_3757_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_3758_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_3759_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_3760_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_3761_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_3762_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_3763_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_3764_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3765_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3766_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3767_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ( C = zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = zero_z2226904508553997617atural ) )
      & ( ( C != zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3768_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_3769_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_3770_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_3771_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_3772_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I2 ) ) ).

% less_mult_imp_div_less
thf(fact_3773_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_3774_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_3775_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_3776_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_3777_div__less__iff__less__mult,axiom,
    ! [Q6: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q6 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q6 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q6 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_3778_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_3779_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_3780_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q6: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q6 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q6 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q6 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3781_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_3782_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_3783_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3784_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_3785_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3786_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3787_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3788_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3789_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3790_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3791_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3792_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3793_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3794_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3795_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3796_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3797_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3798_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3799_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3800_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3801_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3802_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3803_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3804_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3805_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3806_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3807_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3808_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3809_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q7: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q7 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q7 ) ) )
            & ( P @ Q7 ) ) ) ) ).

% split_div'
thf(fact_3810_power__minus__mult,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3811_power__minus__mult,axiom,
    ! [N2: nat,A: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_assn @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3812_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3813_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3814_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_3815_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P7: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P7 @ ( power_8256067586552552935nteger @ P7 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3816_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P7: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P7 @ ( power_power_assn @ P7 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3817_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P7: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P7 @ ( power_power_rat @ P7 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3818_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P7: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P7 @ ( power_power_nat @ P7 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3819_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P7: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P7 @ ( power_power_int @ P7 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3820_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3821_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3822_power__diff__power__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3823_power__diff__power__eq,axiom,
    ! [A: code_natural,N2: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N2 ) )
            = ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N2 ) )
            = ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3824_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3825_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_3826_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_3827_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3828_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3829_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3830_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_3831_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_3832_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_3833_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_3834_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_3835_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_3836_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_3837_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_3838_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_3839_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_3840_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3841_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3842_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3843_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3844_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ W2 ) @ Z ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( times_times_num @ V @ W2 ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3845_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_3846_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_3847_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_3848_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_3849_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( numera5444537566228673987atural @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_3850_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_3851_power__add__numeral,axiom,
    ! [A: assn,M: num,N2: num] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_3852_power__add__numeral,axiom,
    ! [A: rat,M: num,N2: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_3853_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_3854_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_3855_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N2: num,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3856_power__add__numeral2,axiom,
    ! [A: assn,M: num,N2: num,B: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3857_power__add__numeral2,axiom,
    ! [A: rat,M: num,N2: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3858_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3859_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3860_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_3861_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_3862_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_3863_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_3864_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_3865_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3866_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3867_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_3868_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_3869_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_3870_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3871_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_3872_zero__comp__diff__simps_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3873_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3874_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3875_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3876_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3877_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3878_zero__comp__diff__simps_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3879_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3880_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3881_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3882_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3883_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3884_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3885_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3886_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3887_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3888_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3889_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3890_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3891_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3892_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3893_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3894_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3895_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3896_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3897_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3898_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3899_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ ( numera5444537566228673987atural @ V ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ V ) ) @ ( times_2397367101498566445atural @ B @ ( numera5444537566228673987atural @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3900_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3901_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3902_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3903_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3904_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3905_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3906_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_3907_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_3908_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_3909_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
thf(fact_3910_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_3911_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_3912_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_3913_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
thf(fact_3914_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3915_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3916_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3917_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3918_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3919_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3920_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3921_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3922_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_3923_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_3924_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_3925_of__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% of_nat_0
thf(fact_3926_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3927_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3928_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3929_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3930_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3931_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3932_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3933_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3934_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3935_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3936_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3937_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3938_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3939_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3940_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3941_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3942_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3943_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3944_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3945_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3946_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3947_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3948_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3949_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3950_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_3951_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_3952_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3953_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_3954_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3955_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_add
thf(fact_3956_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_3957_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_3958_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_3959_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3960_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3961_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3962_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N2 ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3963_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3964_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_3965_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3966_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3967_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3968_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ N2 )
        = one_one_Code_integer )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3969_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3970_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3971_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3972_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3973_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3974_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3975_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3976_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_3977_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_3978_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3979_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3980_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3981_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_3982_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_3983_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_3984_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3985_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3986_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_3987_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_3988_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3989_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3990_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3991_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3992_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3993_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3994_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3995_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_3996_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_3997_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_3998_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3999_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_4000_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_4001_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_4002_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_4003_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_4004_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_4005_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_4006_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4007_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4008_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4009_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4010_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4011_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4012_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4013_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4014_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4015_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4016_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4017_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4018_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4019_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4020_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4021_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4022_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4023_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_le5570908160329646204atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4024_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4025_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4026_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4027_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4028_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N2 ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4029_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4030_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4031_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4032_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_le1926595141338095240atural @ ( semiri3763490453095760265atural @ X ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4033_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4034_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4035_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4036_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4037_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ I2 ) @ N2 ) @ ( semiri3763490453095760265atural @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4038_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I2 ) @ N2 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4039_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4040_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4041_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4042_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P3: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P3 )
       => ( ( ord_less_nat @ M @ P3 )
         => ( ! [N5: nat] :
                ( ( ord_less_nat @ N5 @ P3 )
               => ( ( P @ N5 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N5 ) @ P3 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_4043_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_4044_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_4045_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_4046_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_4047_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_4048_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_4049_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I2: nat] :
      ( ( P @ K )
     => ( ! [N5: nat] :
            ( ( P @ ( suc @ N5 ) )
           => ( P @ N5 ) )
       => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_4050_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).

% diff_commute
thf(fact_4051_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_4052_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_4053_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_4054_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_4055_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_4056_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_4057_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_4058_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( suc @ N2 )
        = M )
     => ( N2
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_4059_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_4060_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_4061_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_4062_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_4063_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N2 ) )
     != zero_z3403309356797280102nteger ) ).

% of_nat_neq_0
thf(fact_4064_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4065_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4066_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4067_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4068_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4069_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4070_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4071_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4072_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4073_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4074_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4075_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4076_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4077_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4078_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4079_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4080_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_4081_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_4082_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4083_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4084_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4085_mod__mult__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4086_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4087_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4088_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B2 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4089_mod__mult__cong,axiom,
    ! [A: code_natural,C: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C )
        = ( modulo8411746178871703098atural @ A2 @ C ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C )
          = ( modulo8411746178871703098atural @ B2 @ C ) )
       => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
          = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4090_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4091_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4092_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4093_mod__mult__mult2,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
      = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4094_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4095_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4096_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4097_mult__mod__right,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4098_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4099_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4100_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4101_mod__mult__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4102_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4103_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4104_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4105_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4106_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D2 @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_4107_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_4108_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_4109_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_4110_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_4111_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_4112_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4113_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4114_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_4115_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_4116_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_4117_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
      = ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_4118_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D2 @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_4119_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_4120_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_4121_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_4122_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4123_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4124_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4125_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4126_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4127_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4128_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4129_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4130_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ C ) @ A )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4131_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4132_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4133_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4134_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4135_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4136_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4137_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4138_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4139_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4140_inf__period_I1_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X7: code_integer,K4: code_integer] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) )
              & ( Q @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4141_inf__period_I1_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X7: rat,K4: rat] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4142_inf__period_I1_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X7: int,K4: int] :
            ( ( ( P @ X7 )
              & ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4143_inf__period_I2_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X7: code_integer,K4: code_integer] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) )
              | ( Q @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4144_inf__period_I2_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X7: rat,K4: rat] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4145_inf__period_I2_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X7: int,K4: int] :
            ( ( ( P @ X7 )
              | ( Q @ X7 ) )
            = ( ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4146_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_4147_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4148_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4149_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_4150_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_4151_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_4152_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_4153_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_4154_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_4155_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_4156_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_4157_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N5: nat] :
            ( ( P @ ( suc @ N5 ) )
           => ( P @ N5 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_4158_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_4159_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( P @ N5 )
           => ( P @ ( suc @ N5 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_4160_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_4161_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_4162_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_4163_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_4164_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_4165_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_4166_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_4167_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_4168_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_4169_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_4170_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_4171_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N2 )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_4172_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_4173_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_4174_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N2 )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_4175_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N2 @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_4176_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_4177_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_4178_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_4179_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I2 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_4180_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_4181_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_4182_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_4183_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_4184_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_4185_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_4186_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_4187_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_4188_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_4189_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_4190_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_4191_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_4192_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_4193_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_4194_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_4195_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_4196_Suc__le__D,axiom,
    ! [N2: nat,M8: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M8 )
     => ? [M5: nat] :
          ( M8
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_4197_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_4198_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_4199_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_4200_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 )
             => ( P @ M4 ) )
         => ( P @ N5 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_4201_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ M @ N5 )
             => ( ( P @ N5 )
               => ( P @ ( suc @ N5 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_4202_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [N5: nat] : ( R @ N5 @ ( suc @ N5 ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_4203_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_4204_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_4205_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_4206_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_4207_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_4208_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_4209_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4210_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4211_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4212_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4213_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4214_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4215_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4216_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4217_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: int,A: int,B: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4218_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4219_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4220_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4221_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4222_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4223_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4224_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4225_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4226_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4227_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: int,B: int,C: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4228_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4229_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N2 ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N2 ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4230_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4231_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4232_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4233_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_4234_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_4235_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_4236_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_4237_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_4238_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_4239_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_4240_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_less_as_int
thf(fact_4241_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4242_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4243_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4244_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4245_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4246_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4247_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4248_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_4249_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_4250_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_4251_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_4252_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_4253_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_4254_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_4255_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_4256_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_4257_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D ) ) ) ) ).

% mod_eqE
thf(fact_4258_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% mod_eqE
thf(fact_4259_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A5: code_integer,B5: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_4260_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4261_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_4262_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_4263_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer,B5: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_4264_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B5: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4265_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_4266_add__le__imp__le__diff,axiom,
    ! [I2: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
     => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4267_add__le__imp__le__diff,axiom,
    ! [I2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4268_add__le__imp__le__diff,axiom,
    ! [I2: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
     => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4269_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4270_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4271_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4272_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_4273_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_4274_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_4275_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_4276_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_4277_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_4278_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4279_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4280_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4281_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4282_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4283_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4284_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_4285_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4286_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_4287_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_4288_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_4289_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_4290_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_4291_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_4292_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_4293_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_4294_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I2 ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).

% of_nat_mono
thf(fact_4295_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_4296_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_4297_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_4298_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_4299_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_4300_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_4301_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_4302_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_4303_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4304_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4305_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4306_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_4307_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_4308_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_4309_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_4310_div__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N2 ) ) )
      = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_4311_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_4312_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_4313_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_4314_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_4315_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_4316_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_4317_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_4318_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_4319_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_4320_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_4321_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_4322_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_4323_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_4324_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N2 ) ) ).

% zero_le_numeral
thf(fact_4325_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N2 ) ) ).

% zero_le_numeral
thf(fact_4326_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_4327_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_4328_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_4329_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_4330_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N2 ) @ zero_z2226904508553997617atural ) ).

% not_numeral_le_zero
thf(fact_4331_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_4332_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_4333_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_4334_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_4335_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_4336_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N2 ) ) ).

% zero_less_numeral
thf(fact_4337_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_4338_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N2 ) ) ).

% zero_less_numeral
thf(fact_4339_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_4340_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_4341_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_4342_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_4343_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N2 ) @ zero_z2226904508553997617atural ) ).

% not_numeral_less_zero
thf(fact_4344_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) ) ).

% one_le_numeral
thf(fact_4345_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) ) ).

% one_le_numeral
thf(fact_4346_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_4347_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_4348_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_4349_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_4350_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_4351_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_4352_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_4353_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_4354_lift__Suc__mono__less,axiom,
    ! [F: nat > assn,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_assn @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_assn @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_4355_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_rat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_4356_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_num @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_4357_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_4358_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_4359_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > assn,N2: nat,M: nat] :
      ( ! [N5: nat] : ( ord_less_assn @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_assn @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_4360_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N5: nat] : ( ord_less_rat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_4361_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N5: nat] : ( ord_less_num @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_4362_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_4363_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N5: nat] : ( ord_less_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_4364_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_4365_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_4366_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_4367_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_4368_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_4369_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_set_int @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_4370_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_rat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_4371_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_num @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_4372_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_4373_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N5: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_int @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_4374_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_4375_power__Suc2,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_4376_power__Suc2,axiom,
    ! [A: assn,N2: nat] :
      ( ( power_power_assn @ A @ ( suc @ N2 ) )
      = ( times_times_assn @ ( power_power_assn @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_4377_power__Suc2,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_4378_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_4379_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_4380_power__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_4381_power__Suc,axiom,
    ! [A: assn,N2: nat] :
      ( ( power_power_assn @ A @ ( suc @ N2 ) )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_4382_power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_4383_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_4384_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_4385_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_4386_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M3: nat] :
            ( N2
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_4387_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_4388_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_4389_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_4390_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_4391_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_4392_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_4393_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_4394_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_4395_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N5 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct'
thf(fact_4396_nat__compl__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N5 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct
thf(fact_4397_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_4398_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_4399_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).

% le_diff_conv
thf(fact_4400_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_4401_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_4402_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_4403_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( minus_minus_nat @ J @ I2 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_4404_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_4405_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_4406_dec__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ I2 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I2 @ N5 )
             => ( ( ord_less_nat @ N5 @ J )
               => ( ( P @ N5 )
                 => ( P @ ( suc @ N5 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_4407_inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ J )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I2 @ N5 )
             => ( ( ord_less_nat @ N5 @ J )
               => ( ( P @ ( suc @ N5 ) )
                 => ( P @ N5 ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% inc_induct
thf(fact_4408_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_4409_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_4410_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_4411_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_4412_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_4413_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_4414_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_4415_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q5: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).

% less_natE
thf(fact_4416_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).

% less_add_Suc1
thf(fact_4417_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_4418_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_4419_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_4420_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_4421_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_4422_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_4423_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_4424_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_4425_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_4426_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_4427_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_4428_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4429_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4430_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4431_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4432_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4433_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4434_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4435_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4436_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4437_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4438_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4439_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4440_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4441_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4442_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4443_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4444_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4445_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] :
      ( A
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4446_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4447_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4448_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4449_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4450_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4451_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4452_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4453_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4454_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4455_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4456_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4457_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4458_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4459_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4460_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4461_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4462_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4463_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4464_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4465_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4466_le__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_4467_le__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_4468_le__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_4469_le__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_4470_le__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_4471_le__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_4472_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4473_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4474_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4475_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4476_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4477_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4478_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_4479_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_4480_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_4481_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4482_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4483_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4484_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4485_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_4486_power__inject__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) )
        = ( power_8256067586552552935nteger @ B @ ( suc @ N2 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4487_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
        = ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4488_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4489_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_4490_power__le__imp__le__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) @ ( power_8256067586552552935nteger @ B @ ( suc @ N2 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4491_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4492_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4493_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_4494_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_4495_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_4496_mod__eq__nat1E,axiom,
    ! [M: nat,Q6: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q6 )
        = ( modulo_modulo_nat @ N2 @ Q6 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S7: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q6 @ S7 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_4497_mod__eq__nat2E,axiom,
    ! [M: nat,Q6: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q6 )
        = ( modulo_modulo_nat @ N2 @ Q6 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S7: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q6 @ S7 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_4498_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q5: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q5 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_4499_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A3 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_4500_power__gt1,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_4501_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_4502_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_4503_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_4504_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_4505_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4506_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_4507_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I: nat] :
                ( ( ord_less_eq_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_4508_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_4509_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_4510_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_4511_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( P @ N5 )
               => ( P @ ( suc @ N5 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_4512_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_4513_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4514_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_4515_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4516_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_4517_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4518_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_4519_nat__one__le__power,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_4520_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4521_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4522_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_4523_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_4524_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_4525_power__Suc__le__self,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4526_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4527_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4528_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_4529_power__Suc__less__one,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_4530_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_4531_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_4532_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_4533_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4534_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4535_power__diff,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4536_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4537_power__diff,axiom,
    ! [A: code_natural,N2: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4538_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_4539_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4540_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_4541_div__nat__eqI,axiom,
    ! [N2: nat,Q6: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q6 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q6 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q6 ) ) ) ).

% div_nat_eqI
thf(fact_4542_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4543_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4544_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4545_scaling__mono,axiom,
    ! [U: rat,V: rat,R3: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4546_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_4547_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N5: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N5 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_4548_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N5: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N5 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4549_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_4550_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_4551_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_4552_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I4: code_integer] : ( plus_p5714425477246183910nteger @ I4 @ one_one_Code_integer )
          @ N
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_4553_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
     => ( ! [M5: nat,N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ( P @ N5 @ ( modulo_modulo_nat @ M5 @ N5 ) )
             => ( P @ M5 @ N5 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_4554_option_Osize__gen_I2_J,axiom,
    ! [X: produc3260487557148687353it_nat > nat,X2: produc3260487557148687353it_nat] :
      ( ( size_o1731029420616209814it_nat @ X @ ( some_P7913643980934408916it_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4555_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X2: num] :
      ( ( size_option_num @ X @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4556_option_Osize__gen_I2_J,axiom,
    ! [X: produc8664842809031399944it_nat > nat,X2: produc8664842809031399944it_nat] :
      ( ( size_o29782932136985253it_nat @ X @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4557_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_4558_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_4559_pochhammer__code,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N: nat] :
          ( if_Code_integer @ ( N = zero_zero_nat ) @ one_one_Code_integer
          @ ( set_fo1084959871951514735nteger
            @ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_Code_integer ) ) ) ) ).

% pochhammer_code
thf(fact_4560_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_4561_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4562_Diff__cancel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ A3 )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_4563_Diff__cancel,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ A3 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_4564_Diff__cancel,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ A3 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_4565_Diff__cancel,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ A3 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_4566_empty__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A3 )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_4567_empty__Diff,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_4568_empty__Diff,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_4569_empty__Diff,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_4570_Diff__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = A3 ) ).

% Diff_empty
thf(fact_4571_Diff__empty,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Diff_empty
thf(fact_4572_Diff__empty,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Diff_empty
thf(fact_4573_Diff__empty,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Diff_empty
thf(fact_4574_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ B3 @ A3 ) @ A3 )
      = ( sup_su718114333110466843at_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_4575_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ B3 @ A3 ) @ A3 )
      = ( sup_su3035147773818789531at_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_4576_Un__Diff__cancel2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B3 @ A3 ) @ A3 )
      = ( sup_su5525570899277871387at_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_4577_Un__Diff__cancel2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B3 @ A3 ) @ A3 )
      = ( sup_sup_set_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_4578_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B3 @ A3 ) )
      = ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_4579_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A3 @ ( minus_5060654252129873198at_nat @ B3 @ A3 ) )
      = ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_4580_Un__Diff__cancel,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B3 @ A3 ) )
      = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_4581_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_4582_Diff__eq__empty__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_4583_Diff__eq__empty__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_4584_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_4585_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_4586_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_4587_Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B3 @ A3 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_4588_Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( minus_minus_set_int @ B3 @ A3 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_4589_Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_4590_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_4591_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_4592_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_4593_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_4594_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_4595_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_4596_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A3 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_4597_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_4598_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_4599_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_4600_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_4601_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_4602_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_4603_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_4604_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_4605_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_4606_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_4607_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_4608_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_4609_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( modulo_modulo_int @ I2 @ K )
        = I2 )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_4610_plusinfinity,axiom,
    ! [D2: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_4611_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_4612_decr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: 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 @ K )
         => ! [X7: int] :
              ( ( P @ X7 )
             => ( P @ ( minus_minus_int @ X7 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4613_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q6: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_4614_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q6: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_4615_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_4616_q__pos__lemma,axiom,
    ! [B2: int,Q8: int,R7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q8 ) @ R7 ) )
     => ( ( ord_less_int @ R7 @ B2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ord_less_eq_int @ zero_zero_int @ Q8 ) ) ) ) ).

% q_pos_lemma
thf(fact_4617_zdiv__mono2__lemma,axiom,
    ! [B: int,Q6: int,R3: int,B2: int,Q8: int,R7: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q8 ) @ R7 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q8 ) @ R7 ) )
       => ( ( ord_less_int @ R7 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q6 @ Q8 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_4618_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: 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 @ K )
         => ! [X7: int] :
              ( ( P @ X7 )
             => ( P @ ( plus_plus_int @ X7 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_4619_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q6: int,R3: int,B2: int,Q8: int,R7: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q8 ) @ R7 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q8 ) @ R7 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R7 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q8 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_4620_unique__quotient__lemma,axiom,
    ! [B: int,Q8: int,R7: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q8 ) @ R7 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R7 )
       => ( ( ord_less_int @ R7 @ B )
         => ( ( ord_less_int @ R3 @ B )
           => ( ord_less_eq_int @ Q8 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_4621_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q8: int,R7: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q8 ) @ R7 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( ord_less_int @ B @ R7 )
           => ( ord_less_eq_int @ Q6 @ Q8 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_4622_imp__le__cong,axiom,
    ! [X: int,X10: int,P: $o,P6: $o] :
      ( ( X = X10 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X10 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_4623_conj__le__cong,axiom,
    ! [X: int,X10: int,P: $o,P6: $o] :
      ( ( X = X10 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X10 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_4624_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_4625_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) ) ) ) ).

% split_zdiv
thf(fact_4626_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q6: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( divide_divide_int @ A @ B )
            = Q6 ) ) ) ) ).

% int_div_neg_eq
thf(fact_4627_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q6: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q6 ) ) ) ) ).

% int_div_pos_eq
thf(fact_4628_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_4629_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_4630_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_4631_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_4632_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
        = ( ord_less_eq_int @ K @ I2 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_4633_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_4634_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_4635_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_4636_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_4637_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_4638_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_4639_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( divide_divide_int @ I2 @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_4640_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_4641_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_4642_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_4643_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_4644_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_4645_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_4646_Diff__mono,axiom,
    ! [A3: set_nat,C3: set_nat,D3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ D3 @ B3 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ C3 @ D3 ) ) ) ) ).

% Diff_mono
thf(fact_4647_Diff__mono,axiom,
    ! [A3: set_int,C3: set_int,D3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C3 )
     => ( ( ord_less_eq_set_int @ D3 @ B3 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B3 ) @ ( minus_minus_set_int @ C3 @ D3 ) ) ) ) ).

% Diff_mono
thf(fact_4648_Diff__subset,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 ) ).

% Diff_subset
thf(fact_4649_Diff__subset,axiom,
    ! [A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B3 ) @ A3 ) ).

% Diff_subset
thf(fact_4650_double__diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C3 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_4651_double__diff,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C3 )
       => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C3 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_4652_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_4653_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ C3 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4654_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4655_Diff__Int__distrib,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_4656_Diff__Int__distrib,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A3 ) @ ( inf_inf_set_nat @ C3 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_4657_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_4658_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_4659_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_4660_Diff__Int2,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_4661_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ C3 ) ) ) ).

% Int_Diff
thf(fact_4662_Int__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
      = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C3 ) ) ) ).

% Int_Diff
thf(fact_4663_set__diff__diff__left,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) @ C3 )
      = ( minus_8321449233255521966at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) ) ) ).

% set_diff_diff_left
thf(fact_4664_set__diff__diff__left,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) @ C3 )
      = ( minus_5060654252129873198at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) ) ) ).

% set_diff_diff_left
thf(fact_4665_set__diff__diff__left,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) @ C3 )
      = ( minus_3314409938677909166at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) ) ) ).

% set_diff_diff_left
thf(fact_4666_set__diff__diff__left,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% set_diff_diff_left
thf(fact_4667_Un__Diff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ C3 ) @ ( minus_8321449233255521966at_nat @ B3 @ C3 ) ) ) ).

% Un_Diff
thf(fact_4668_Un__Diff,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A3 @ C3 ) @ ( minus_5060654252129873198at_nat @ B3 @ C3 ) ) ) ).

% Un_Diff
thf(fact_4669_Un__Diff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) @ C3 )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ C3 ) @ ( minus_3314409938677909166at_nat @ B3 @ C3 ) ) ) ).

% Un_Diff
thf(fact_4670_Un__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C3 ) @ ( minus_minus_set_nat @ B3 @ C3 ) ) ) ).

% Un_Diff
thf(fact_4671_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ B3 )
     => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4672_psubset__imp__ex__mem,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_set_o @ A3 @ B3 )
     => ? [B4: $o] : ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4673_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B3 )
     => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4674_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4675_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4676_subset__minus__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
thf(fact_4677_subset__minus__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( minus_minus_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_4678_subset__minus__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( minus_minus_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_4679_subset__minus__empty,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( minus_minus_set_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_4680_disjoint__alt__simp2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_4681_disjoint__alt__simp2,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
thf(fact_4682_disjoint__alt__simp2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
thf(fact_4683_disjoint__alt__simp2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_4684_disjoint__alt__simp1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_4685_disjoint__alt__simp1,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
thf(fact_4686_disjoint__alt__simp1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
thf(fact_4687_disjoint__alt__simp1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_4688_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_4689_Int__Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B3 ) @ ( minus_minus_set_o @ A3 @ B3 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_4690_Int__Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ ( minus_minus_set_int @ A3 @ B3 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_4691_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_4692_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_4693_Diff__triv,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_4694_Diff__triv,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_4695_Diff__triv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_4696_Diff__partition,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ B3 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_4697_Diff__partition,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A3 @ B3 )
     => ( ( sup_su3035147773818789531at_nat @ A3 @ ( minus_5060654252129873198at_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_4698_Diff__partition,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ B3 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_4699_Diff__partition,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_4700_Diff__partition,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( sup_sup_set_int @ A3 @ ( minus_minus_set_int @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_4701_Diff__subset__conv,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) @ C3 )
      = ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_4702_Diff__subset__conv,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) @ C3 )
      = ( ord_le8081472938463900775at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_4703_Diff__subset__conv,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) @ C3 )
      = ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_4704_Diff__subset__conv,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_4705_Diff__subset__conv,axiom,
    ! [A3: set_int,B3: set_int,C3: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B3 ) @ C3 )
      = ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_4706_pochhammer__pos,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( comm_s8582702949713902594nteger @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_4707_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_4708_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_4709_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_4710_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_4711_Un__Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_4712_Un__Diff__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_4713_Un__Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_4714_Un__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_4715_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_4716_Int__Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_4717_Int__Diff__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_4718_Int__Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_4719_Int__Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_4720_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_4721_Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B3 @ C3 ) )
      = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) @ ( minus_8321449233255521966at_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_4722_Diff__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ A3 @ ( inf_in1697001100524423349at_nat @ B3 @ C3 ) )
      = ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) @ ( minus_5060654252129873198at_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_4723_Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B3 @ C3 ) )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) @ ( minus_3314409938677909166at_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_4724_Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_4725_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_4726_Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,C3: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B3 @ C3 ) )
      = ( inf_in4302113700860409141at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) @ ( minus_8321449233255521966at_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_4727_Diff__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ A3 @ ( sup_su3035147773818789531at_nat @ B3 @ C3 ) )
      = ( inf_in1697001100524423349at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) @ ( minus_5060654252129873198at_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_4728_Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B3 @ C3 ) )
      = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) @ ( minus_3314409938677909166at_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_4729_Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_4730_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_4731_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_4732_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_4733_pochhammer__nonneg,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( comm_s8582702949713902594nteger @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_4734_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_4735_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_4736_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_4737_disjoint__alt__simp3,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_4738_disjoint__alt__simp3,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
thf(fact_4739_disjoint__alt__simp3,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
thf(fact_4740_disjoint__alt__simp3,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_4741_pochhammer__rec,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ A @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4742_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4743_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4744_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4745_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4746_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4747_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4748_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ ( comm_s8582702949713902594nteger @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4749_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4750_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4751_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4752_pochhammer__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N2 ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4753_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4754_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4755_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4756_pochhammer__product_H,axiom,
    ! [Z: code_integer,N2: nat,M: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4757_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4758_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4759_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4760_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s8582702949713902594nteger @ Z @ N2 )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4761_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N5: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N5 ) ) ).

% real_arch_simple
thf(fact_4762_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N5: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N5 ) ) ).

% reals_Archimedean2
thf(fact_4763_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% zle_diff1_eq
thf(fact_4764_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% zle_add1_eq_le
thf(fact_4765_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N5: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N5 )
          & ( K
            = ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_4766_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N5: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% pos_int_cases
thf(fact_4767_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_4768_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N5: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N5 ) ) ) ).

% nonneg_int_cases
thf(fact_4769_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N5: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N5 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_4770_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_4771_Diff__idemp,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ B3 )
      = ( minus_minus_set_nat @ A3 @ B3 ) ) ).

% Diff_idemp
thf(fact_4772_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( ( member8440522571783428010at_nat @ C @ A3 )
        & ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_4773_Diff__iff,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B3 ) )
      = ( ( member_o @ C @ A3 )
        & ~ ( member_o @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_4774_Diff__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) )
      = ( ( member_set_nat @ C @ A3 )
        & ~ ( member_set_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_4775_Diff__iff,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B3 ) )
      = ( ( member_int @ C @ A3 )
        & ~ ( member_int @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_4776_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( ( member_nat @ C @ A3 )
        & ~ ( member_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_4777_DiffI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ C @ B3 )
       => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_4778_DiffI,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ A3 )
     => ( ~ ( member_o @ C @ B3 )
       => ( member_o @ C @ ( minus_minus_set_o @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_4779_DiffI,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( ~ ( member_set_nat @ C @ B3 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_4780_DiffI,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ A3 )
     => ( ~ ( member_int @ C @ B3 )
       => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_4781_DiffI,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_4782_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4783_minus__set__def,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( minus_minus_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A6 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4784_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A6 )
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4785_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4786_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4787_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 )
            @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4788_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_4789_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
     => ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_4790_DiffD2,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B3 ) )
     => ~ ( member_o @ C @ B3 ) ) ).

% DiffD2
thf(fact_4791_DiffD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) )
     => ~ ( member_set_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_4792_DiffD2,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B3 ) )
     => ~ ( member_int @ C @ B3 ) ) ).

% DiffD2
thf(fact_4793_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ~ ( member_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_4794_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_4795_DiffD1,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B3 ) )
     => ( member_o @ C @ A3 ) ) ).

% DiffD1
thf(fact_4796_DiffD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) )
     => ( member_set_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_4797_DiffD1,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B3 ) )
     => ( member_int @ C @ A3 ) ) ).

% DiffD1
thf(fact_4798_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ( member_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_4799_DiffE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
         => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_4800_DiffE,axiom,
    ! [C: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B3 ) )
     => ~ ( ( member_o @ C @ A3 )
         => ( member_o @ C @ B3 ) ) ) ).

% DiffE
thf(fact_4801_DiffE,axiom,
    ! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) )
     => ~ ( ( member_set_nat @ C @ A3 )
         => ( member_set_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_4802_DiffE,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B3 ) )
     => ~ ( ( member_int @ C @ A3 )
         => ( member_int @ C @ B3 ) ) ) ).

% DiffE
thf(fact_4803_DiffE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ( member_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_4804_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              & ~ ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4805_set__diff__eq,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A6 )
              & ~ ( member_o @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4806_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X4 @ A6 )
              & ~ ( member5262025264175285858nt_int @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4807_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A6 )
              & ~ ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4808_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A6 )
              & ~ ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4809_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A6 )
              & ~ ( member_int @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4810_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              & ~ ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4811_int__ge__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I2 )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_ge_induct
thf(fact_4812_int__le__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I2 @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_le_induct
thf(fact_4813_int__induct,axiom,
    ! [P: int > $o,K: int,I2: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_induct
thf(fact_4814_int__gr__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_int @ K @ I2 )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_gr_induct
thf(fact_4815_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z )
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
thf(fact_4816_int__less__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I2 @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_less_induct
thf(fact_4817_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_4818_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_4819_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_4820_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_4821_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_4822_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
      = ( ord_less_int @ W2 @ Z ) ) ).

% add1_zle_eq
thf(fact_4823_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_4824_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_4825_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_4826_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_4827_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_4828_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_4829_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q6 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q6 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q6 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4830_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4831_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4832_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4833_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: assn] :
      ( ( uminus_uminus_assn @ ( uminus_uminus_assn @ X ) )
      = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_4834_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: product_unit] :
      ( ( uminus2952777764628376836t_unit @ ( uminus2952777764628376836t_unit @ X ) )
      = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_4835_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( uminus_uminus_assn @ X )
        = ( uminus_uminus_assn @ Y ) )
      = ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_4836_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( uminus2952777764628376836t_unit @ X )
        = ( uminus2952777764628376836t_unit @ Y ) )
      = ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_4837_Compl__anti__mono,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% Compl_anti_mono
thf(fact_4838_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B3 ) )
      = ( ord_less_eq_set_int @ B3 @ A3 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4839_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4840_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4841_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4842_compl__le__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_eq_assn @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4843_compl__le__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) )
      = ( ord_le3221252021190050221t_unit @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4844_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4845_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4846_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4847_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4848_compl__less__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) )
      = ( ord_less_assn @ Y @ X ) ) ).

% compl_less_compl_iff
thf(fact_4849_compl__less__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) )
      = ( ord_le361264281704409273t_unit @ Y @ X ) ) ).

% compl_less_compl_iff
thf(fact_4850_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4851_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4852_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4853_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4854_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4855_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4856_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4857_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4858_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4859_Compl__disjoint2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ A3 )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_4860_Compl__disjoint2,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_4861_Compl__disjoint2,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_4862_Compl__disjoint2,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ A3 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_4863_Compl__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_4864_Compl__disjoint,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_4865_Compl__disjoint,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_4866_Compl__disjoint,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_4867_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4868_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4869_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4870_inter__compl__diff__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) )
      = ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) ).

% inter_compl_diff_conv
thf(fact_4871_inter__compl__diff__conv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) )
      = ( minus_minus_set_nat @ A3 @ B3 ) ) ).

% inter_compl_diff_conv
thf(fact_4872_Diff__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Diff_Compl
thf(fact_4873_Diff__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Diff_Compl
thf(fact_4874_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_4875_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_4876_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_4877_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B3 ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_4878_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( minus_5060654252129873198at_nat @ A3 @ B3 ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_4879_Compl__Diff__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B3 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_4880_Compl__Diff__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ B3 ) ) ).

% Compl_Diff_eq
thf(fact_4881_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_4882_mod__not__dist,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P ) @ H )
      = ( ( in_range @ H )
        & ~ ( rep_assn @ P @ H ) ) ) ).

% mod_not_dist
thf(fact_4883_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4884_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4885_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4886_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4887_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_4888_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_4889_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4890_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4891_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4892_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_4893_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_4894_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_4895_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4896_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4897_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4898_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_4899_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_4900_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_4901_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_4902_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4903_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_4904_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_4905_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4906_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_4907_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_4908_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_4909_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_4910_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4911_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_4912_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4913_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4914_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4915_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4916_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_4917_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_4918_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_4919_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_4920_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_4921_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_4922_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_4923_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_4924_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_4925_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ X ) )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4926_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4927_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4928_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4929_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4930_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_cancel_right
thf(fact_4931_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4932_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4933_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4934_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4935_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4936_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_cancel_left
thf(fact_4937_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ ( uminus6524753893492686040at_nat @ X ) ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
thf(fact_4938_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_right
thf(fact_4939_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_4940_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_4941_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
      = bot_bot_assn ) ).

% inf_compl_bot_right
thf(fact_4942_inf__compl__bot__right,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) ) )
      = bot_bot_Product_unit ) ).

% inf_compl_bot_right
thf(fact_4943_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
thf(fact_4944_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left2
thf(fact_4945_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_4946_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_4947_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left2
thf(fact_4948_inf__compl__bot__left2,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
      = bot_bot_Product_unit ) ).

% inf_compl_bot_left2
thf(fact_4949_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
thf(fact_4950_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left1
thf(fact_4951_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_4952_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_4953_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left1
thf(fact_4954_inf__compl__bot__left1,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( inf_inf_Product_unit @ X @ Y ) )
      = bot_bot_Product_unit ) ).

% inf_compl_bot_left1
thf(fact_4955_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4956_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( sup_su718114333110466843at_nat @ X @ Y ) )
      = ( inf_in4302113700860409141at_nat @ ( uminus4384627049435823934at_nat @ X ) @ ( uminus4384627049435823934at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4957_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
      = ( inf_in1697001100524423349at_nat @ ( uminus2330091110623919550at_nat @ X ) @ ( uminus2330091110623919550at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4958_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( uminus935396558254630718at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4959_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4960_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: assn,Y: assn] :
      ( ( uminus_uminus_assn @ ( sup_sup_assn @ X @ Y ) )
      = ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4961_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( uminus2952777764628376836t_unit @ ( sup_sup_Product_unit @ X @ Y ) )
      = ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4962_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4963_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ X ) @ ( uminus4384627049435823934at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4964_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ ( uminus2330091110623919550at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4965_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( uminus935396558254630718at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4966_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4967_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: assn,Y: assn] :
      ( ( uminus_uminus_assn @ ( inf_inf_assn @ X @ Y ) )
      = ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4968_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( uminus2952777764628376836t_unit @ ( inf_inf_Product_unit @ X @ Y ) )
      = ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4969_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_4970_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4971_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4972_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4973_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_4974_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_4975_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4976_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4977_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4978_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_4979_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_4980_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_4981_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4982_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4983_semiring__norm_I172_J,axiom,
    ! [V: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4984_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4985_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4986_semiring__norm_I171_J,axiom,
    ! [V: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4987_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4988_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4989_semiring__norm_I170_J,axiom,
    ! [V: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4990_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4991_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4992_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4993_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4994_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4995_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4996_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4997_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4998_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4999_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_5000_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5001_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5002_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5003_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5004_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5005_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5006_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5007_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5008_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5009_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5010_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5011_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5012_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5013_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
      = ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5014_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
      = ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5015_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I3: int] :
          ( X
         != ( product_Pair_int_int @ D @ I3 ) ) ).

% small_lazy'.cases
thf(fact_5016_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D: int,I3: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_5017_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I3: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_5018_unique__remainder,axiom,
    ! [A: int,B: int,Q6: int,R3: int,Q8: int,R7: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q8 @ R7 ) )
       => ( R3 = R7 ) ) ) ).

% unique_remainder
thf(fact_5019_unique__quotient,axiom,
    ! [A: int,B: int,Q6: int,R3: int,Q8: int,R7: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q8 @ R7 ) )
       => ( Q6 = Q8 ) ) ) ).

% unique_quotient
thf(fact_5020_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5021_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_5022_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_5023_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5024_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5025_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5026_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5027_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5028_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5029_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5030_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5031_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5032_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5033_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5034_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5035_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5036_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5037_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5038_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5039_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5040_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5041_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5042_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_5043_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_5044_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5045_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5046_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5047_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5048_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5049_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5050_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_5051_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5052_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_5053_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).

% zabs_def
thf(fact_5054_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_5055_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_5056_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_5057_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5058_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5059_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5060_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_5061_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5062_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_5063_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5064_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5065_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5066_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5067_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5068_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5069_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5070_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5071_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_5072_compl__le__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ X )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5073_compl__le__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ X )
     => ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5074_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5075_compl__le__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ ( uminus_uminus_assn @ X ) )
     => ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5076_compl__le__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) )
     => ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5077_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
     => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5078_compl__mono,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ord_less_eq_assn @ ( uminus_uminus_assn @ Y ) @ ( uminus_uminus_assn @ X ) ) ) ).

% compl_mono
thf(fact_5079_compl__mono,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ ( uminus2952777764628376836t_unit @ X ) ) ) ).

% compl_mono
thf(fact_5080_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).

% compl_mono
thf(fact_5081_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5082_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5083_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5084_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_5085_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5086_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5087_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5088_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5089_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5090_compl__less__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_assn @ Y @ ( uminus_uminus_assn @ X ) )
     => ( ord_less_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% compl_less_swap1
thf(fact_5091_compl__less__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le361264281704409273t_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) )
     => ( ord_le361264281704409273t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).

% compl_less_swap1
thf(fact_5092_compl__less__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_assn @ ( uminus_uminus_assn @ Y ) @ X )
     => ( ord_less_assn @ ( uminus_uminus_assn @ X ) @ Y ) ) ).

% compl_less_swap2
thf(fact_5093_compl__less__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ X )
     => ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) ) ).

% compl_less_swap2
thf(fact_5094_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5095_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5096_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5097_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_5098_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_5099_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_5100_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A5: assn,B5: assn] : ( inf_inf_assn @ A5 @ ( uminus_uminus_assn @ B5 ) ) ) ) ).

% minus_assn_def
thf(fact_5101_Collect__imp__eq,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su6024340866399070445nt_int @ ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5102_Collect__imp__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5103_Collect__imp__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5104_Collect__imp__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5105_Collect__imp__eq,axiom,
    ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ ( collec7088162979684241874at_nat @ P ) ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5106_Collect__imp__eq,axiom,
    ! [P: produc4166570645942440679at_nat > $o,Q: produc4166570645942440679at_nat > $o] :
      ( ( collec5204685387357076818at_nat
        @ ^ [X4: produc4166570645942440679at_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ ( collec5204685387357076818at_nat @ P ) ) @ ( collec5204685387357076818at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5107_Collect__imp__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5108_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5109_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_5110_div__int__unique,axiom,
    ! [K: int,L: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q6 @ R3 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q6 ) ) ).

% div_int_unique
thf(fact_5111_zminus1__lemma,axiom,
    ! [A: int,B: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q6 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q6 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5112_mod__int__unique,axiom,
    ! [K: int,L: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q6 @ R3 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R3 ) ) ).

% mod_int_unique
thf(fact_5113_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5114_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5115_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5116_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5117_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5118_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5119_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5120_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5121_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5122_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5123_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5124_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5125_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5126_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5127_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5128_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5129_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5130_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5131_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5132_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5133_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5134_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5135_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D2 )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5136_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D2 )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5137_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5138_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5139_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5140_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5141_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5142_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5143_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_5144_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_5145_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% le_minus_one_simps(2)
thf(fact_5146_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_5147_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_5148_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(4)
thf(fact_5149_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5150_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5151_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5152_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5153_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5154_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5155_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_5156_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_5157_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_5158_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_5159_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_5160_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_5161_numeral__times__minus__swap,axiom,
    ! [W2: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W2 ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5162_numeral__times__minus__swap,axiom,
    ! [W2: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5163_numeral__times__minus__swap,axiom,
    ! [W2: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5164_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5165_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5166_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5167_diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ X4 @ ( uminus6524753893492686040at_nat @ Y4 ) ) ) ) ).

% diff_eq
thf(fact_5168_diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y4 ) ) ) ) ).

% diff_eq
thf(fact_5169_diff__eq,axiom,
    ( minus_minus_assn
    = ( ^ [X4: assn,Y4: assn] : ( inf_inf_assn @ X4 @ ( uminus_uminus_assn @ Y4 ) ) ) ) ).

% diff_eq
thf(fact_5170_diff__eq,axiom,
    ( minus_3524152463667985524t_unit
    = ( ^ [X4: product_unit,Y4: product_unit] : ( inf_inf_Product_unit @ X4 @ ( uminus2952777764628376836t_unit @ Y4 ) ) ) ) ).

% diff_eq
thf(fact_5171_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ A ) @ ( inf_in2572325071724192079at_nat @ X @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
thf(fact_5172_inf__cancel__left2,axiom,
    ! [X: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( inf_inf_set_o @ X @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left2
thf(fact_5173_inf__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( inf_inf_set_nat @ X @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_5174_inf__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A ) @ ( inf_inf_set_int @ X @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_5175_inf__cancel__left2,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A ) @ ( inf_inf_assn @ X @ B ) )
      = bot_bot_assn ) ).

% inf_cancel_left2
thf(fact_5176_inf__cancel__left2,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A ) @ ( inf_inf_Product_unit @ X @ B ) )
      = bot_bot_Product_unit ) ).

% inf_cancel_left2
thf(fact_5177_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ A ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
thf(fact_5178_inf__cancel__left1,axiom,
    ! [X: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left1
thf(fact_5179_inf__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_5180_inf__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_5181_inf__cancel__left1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ A ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B ) )
      = bot_bot_assn ) ).

% inf_cancel_left1
thf(fact_5182_inf__cancel__left1,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ A ) @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
      = bot_bot_Product_unit ) ).

% inf_cancel_left1
thf(fact_5183_subset__Compl__self__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
      = ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_5184_subset__Compl__self__eq,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_5185_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_5186_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_5187_Compl__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_5188_Compl__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_5189_Compl__Int,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A3 ) @ ( uminus2330091110623919550at_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_5190_Compl__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_5191_Compl__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_5192_Compl__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_5193_Compl__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
      = ( inf_in4302113700860409141at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_5194_Compl__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
      = ( inf_in1697001100524423349at_nat @ ( uminus2330091110623919550at_nat @ A3 ) @ ( uminus2330091110623919550at_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_5195_Compl__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
      = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_5196_Compl__Un,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_5197_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_5198_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5199_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5200_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5201_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5202_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5203_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5204_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5205_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5206_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5207_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5208_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5209_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5210_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5211_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R3 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5212_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R3 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R3 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5213_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R3 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R3 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5214_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R3 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5215_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R3 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5216_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R3 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R3 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5217_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5218_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5219_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5220_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5221_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5222_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5223_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5224_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5225_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_5226_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_5227_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5228_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_5229_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% le_minus_one_simps(1)
thf(fact_5230_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5231_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_5232_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(3)
thf(fact_5233_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5234_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5235_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5236_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_5237_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5238_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_5239_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_5240_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5241_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_5242_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_5243_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5244_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_5245_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5246_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5247_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5248_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5249_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_5250_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_5251_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5252_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_5253_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_5254_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5255_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_5256_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_5257_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5258_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_5259_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_5260_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5261_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5262_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5263_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5264_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5265_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5266_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_5267_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5268_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_5269_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_5270_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5271_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_5272_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_5273_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5274_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_5275_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5276_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5277_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5278_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5279_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_5280_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).

% inf_shunt
thf(fact_5281_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_5282_inf__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% inf_shunt
thf(fact_5283_inf__shunt,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( inf_inf_Product_unit @ X @ Y )
        = bot_bot_Product_unit )
      = ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).

% inf_shunt
thf(fact_5284_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% inf_shunt
thf(fact_5285_sup__neg__inf,axiom,
    ! [P3: set_Pr1261947904930325089at_nat,Q6: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P3 @ ( sup_su6327502436637775413at_nat @ Q6 @ R3 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P3 @ ( uminus6524753893492686040at_nat @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5286_sup__neg__inf,axiom,
    ! [P3: set_Pr8693737435421807431at_nat,Q6: set_Pr8693737435421807431at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ P3 @ ( sup_su718114333110466843at_nat @ Q6 @ R3 ) )
      = ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ P3 @ ( uminus4384627049435823934at_nat @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5287_sup__neg__inf,axiom,
    ! [P3: set_Pr8551490117392284871at_nat,Q6: set_Pr8551490117392284871at_nat,R3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ P3 @ ( sup_su3035147773818789531at_nat @ Q6 @ R3 ) )
      = ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ P3 @ ( uminus2330091110623919550at_nat @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5288_sup__neg__inf,axiom,
    ! [P3: set_Pr4329608150637261639at_nat,Q6: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ P3 @ ( sup_su5525570899277871387at_nat @ Q6 @ R3 ) )
      = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P3 @ ( uminus935396558254630718at_nat @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5289_sup__neg__inf,axiom,
    ! [P3: set_nat,Q6: set_nat,R3: set_nat] :
      ( ( ord_less_eq_set_nat @ P3 @ ( sup_sup_set_nat @ Q6 @ R3 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P3 @ ( uminus5710092332889474511et_nat @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5290_sup__neg__inf,axiom,
    ! [P3: assn,Q6: assn,R3: assn] :
      ( ( ord_less_eq_assn @ P3 @ ( sup_sup_assn @ Q6 @ R3 ) )
      = ( ord_less_eq_assn @ ( inf_inf_assn @ P3 @ ( uminus_uminus_assn @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5291_sup__neg__inf,axiom,
    ! [P3: product_unit,Q6: product_unit,R3: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ P3 @ ( sup_sup_Product_unit @ Q6 @ R3 ) )
      = ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ P3 @ ( uminus2952777764628376836t_unit @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5292_sup__neg__inf,axiom,
    ! [P3: set_int,Q6: set_int,R3: set_int] :
      ( ( ord_less_eq_set_int @ P3 @ ( sup_sup_set_int @ Q6 @ R3 ) )
      = ( ord_less_eq_set_int @ ( inf_inf_set_int @ P3 @ ( uminus1532241313380277803et_int @ Q6 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_5293_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5294_shunt2,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X @ ( uminus4384627049435823934at_nat @ Y ) ) @ Z )
      = ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5295_shunt2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ ( uminus2330091110623919550at_nat @ Y ) ) @ Z )
      = ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5296_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ ( uminus935396558254630718at_nat @ Y ) ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5297_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5298_shunt2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ ( uminus_uminus_assn @ Y ) ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5299_shunt2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) @ Z )
      = ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5300_shunt2,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) @ Z )
      = ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) ) ) ).

% shunt2
thf(fact_5301_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5302_shunt1,axiom,
    ! [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X @ Y ) @ Z )
      = ( ord_le3000389064537975527at_nat @ X @ ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5303_shunt1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ Z )
      = ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5304_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5305_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5306_shunt1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ ( uminus_uminus_assn @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5307_shunt1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
      = ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5308_shunt1,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ Z )
      = ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_5309_power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5310_power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5311_power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5312_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q6: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q6 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q6 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5313_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_5314_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_5315_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_5316_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_5317_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_5318_int__cases4,axiom,
    ! [M: int] :
      ( ! [N5: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N5 ) )
     => ~ ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) ) ) ) ).

% int_cases4
thf(fact_5319_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_5320_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_5321_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N5: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).

% nonpos_int_cases
thf(fact_5322_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_5323_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5324_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5325_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5326_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5327_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5328_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5329_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5330_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5331_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5332_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5333_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5334_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5335_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5336_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5337_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5338_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5339_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P4: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( in_range @ H6 )
              & ~ ( rep_assn @ P4 @ H6 ) ) ) ) ) ).

% uminus_assn_def
thf(fact_5340_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
       => ~ ! [N5: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ) ).

% int_cases3
thf(fact_5341_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_5342_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_5343_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_5344_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_5345_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_5346_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_5347_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_5348_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_5349_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_5350_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_5351_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_5352_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N5: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N5 ) ) ) ) ) ).

% negD
thf(fact_5353_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_5354_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A3 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5355_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_5356_power__minus_H,axiom,
    ! [X: int,N2: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N2 )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_5357_power__minus_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N2 )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_5358_power__minus_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N2 )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_5359_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5360_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5361_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5362_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5363_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5364_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5365_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5366_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5367_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N5: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% neg_int_cases
thf(fact_5368_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5369_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_5370_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5371_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5372_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_5373_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_5374_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_5375_pochhammer__absorb__comp,axiom,
    ! [R3: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K ) )
      = ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5376_pochhammer__absorb__comp,axiom,
    ! [R3: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K ) )
      = ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5377_pochhammer__absorb__comp,axiom,
    ! [R3: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R3 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R3 ) @ K ) )
      = ( times_3573771949741848930nteger @ R3 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5378_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_5379_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_5380_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5381_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5382_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5383_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5384_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5385_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5386_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_5387_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5388_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_5389_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5390_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_5391_ComplI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ C @ A3 )
     => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) ) ) ).

% ComplI
thf(fact_5392_ComplI,axiom,
    ! [C: $o,A3: set_o] :
      ( ~ ( member_o @ C @ A3 )
     => ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) ) ) ).

% ComplI
thf(fact_5393_ComplI,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A3 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_5394_ComplI,axiom,
    ! [C: nat,A3: set_nat] :
      ( ~ ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_5395_ComplI,axiom,
    ! [C: int,A3: set_int] :
      ( ~ ( member_int @ C @ A3 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% ComplI
thf(fact_5396_Compl__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) )
      = ( ~ ( member8440522571783428010at_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_5397_Compl__iff,axiom,
    ! [C: $o,A3: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
      = ( ~ ( member_o @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_5398_Compl__iff,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
      = ( ~ ( member_set_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_5399_Compl__iff,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( ~ ( member_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_5400_Compl__iff,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( ~ ( member_int @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_5401_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_5402_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_5403_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5404_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_5405_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_5406_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_5407_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_5408_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_5409_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_5410_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_5411_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_5412_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_5413_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5414_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_5415_Collect__neg__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X4: product_prod_int_int] :
            ~ ( P @ X4 ) )
      = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_5416_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ~ ( P @ X4 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_5417_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ~ ( P @ X4 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_5418_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ~ ( P @ X4 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_5419_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ~ ( P @ X4 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_5420_Compl__eq,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ~ ( member8440522571783428010at_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5421_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ~ ( member_o @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5422_Compl__eq,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ~ ( member5262025264175285858nt_int @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5423_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ~ ( member_list_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5424_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ~ ( member_set_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5425_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ~ ( member_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5426_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ~ ( member_int @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5427_ComplD,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A3 ) )
     => ~ ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_5428_ComplD,axiom,
    ! [C: $o,A3: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
     => ~ ( member_o @ C @ A3 ) ) ).

% ComplD
thf(fact_5429_ComplD,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
     => ~ ( member_set_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_5430_ComplD,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
     => ~ ( member_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_5431_ComplD,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
     => ~ ( member_int @ C @ A3 ) ) ).

% ComplD
thf(fact_5432_uminus__set__def,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( uminus8676089048583255045_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5433_uminus__set__def,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ( uminus_uminus_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5434_uminus__set__def,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( uminus7117520113953359693_int_o
            @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5435_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5436_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5437_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5438_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5439_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_5440_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_5441_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_5442_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_5443_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_5444_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5445_option_Osize_I4_J,axiom,
    ! [X2: produc3260487557148687353it_nat] :
      ( ( size_s2363601347547812957it_nat @ ( some_P7913643980934408916it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5446_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5447_option_Osize_I4_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( ( size_s8766407808098229740it_nat @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_5448_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5449_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5450_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5451_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5452_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5453_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5454_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5455_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5456_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5457_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5458_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5459_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5460_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_5461_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_5462_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_5463_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_5464_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_5465_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_5466_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_5467_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_18347121197199848620nteger @ ( times_times_int @ W2 @ Z ) )
      = ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W2 ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_mult
thf(fact_5468_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_5469_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_5470_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_5471_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( ( nat2 @ I2 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_5472_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W2 @ Z ) ) ) ).

% zless_nat_conj
thf(fact_5473_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_5474_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5475_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5476_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5477_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5478_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5479_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5480_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5481_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5482_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5483_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5484_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5485_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5486_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5487_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5488_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5489_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5490_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5491_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5492_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5493_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5494_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5495_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5496_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5497_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5498_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5499_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5500_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5501_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5502_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5503_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5504_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5505_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5506_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5507_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5508_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5509_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5510_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_5511_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_5512_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_5513_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_5514_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_5515_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_5516_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_5517_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_5518_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_5519_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_5520_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_5521_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z ) )
        = ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_nat_nat
thf(fact_5522_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5523_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5524_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5525_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5526_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5527_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5528_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5529_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5530_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_5531_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_5532_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_5533_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_5534_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_5535_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5536_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5537_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5538_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5539_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5540_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5541_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5542_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5543_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5544_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5545_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5546_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5547_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_5548_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_5549_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_5550_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_5551_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_5552_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_5553_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5554_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5555_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_5556_mult__of__int__commute,axiom,
    ! [X: int,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
      = ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5557_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5558_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5559_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_5560_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs2 ) )
             => ( P @ Ys ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_5561_of__nat__floor,axiom,
    ! [R3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_5562_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_5563_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_5564_floor__split,axiom,
    ! [P: int > $o,T5: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T5 ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T5 )
              & ( ord_less_rat @ T5 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_5565_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_5566_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_5567_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_5568_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_5569_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_5570_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_5571_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_5572_all__nat,axiom,
    ( ( ^ [P5: nat > $o] :
        ! [X5: nat] : ( P5 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).

% all_nat
thf(fact_5573_ex__nat,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X5: nat] : ( P5 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ? [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
          & ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).

% ex_nat
thf(fact_5574_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_5575_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_5576_floor__divide__lower,axiom,
    ! [Q6: rat,P3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q6 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P3 @ Q6 ) ) ) @ Q6 ) @ P3 ) ) ).

% floor_divide_lower
thf(fact_5577_of__int__of__nat,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_5578_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_5579_of__int__of__nat,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_5580_floor__divide__upper,axiom,
    ! [Q6: rat,P3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q6 )
     => ( ord_less_rat @ P3 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P3 @ Q6 ) ) ) @ one_one_rat ) @ Q6 ) ) ) ).

% floor_divide_upper
thf(fact_5581_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_mono_iff
thf(fact_5582_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_5583_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_5584_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_5585_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_5586_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_5587_of__nat__ceiling,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ R3 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5588_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5589_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5590_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_5591_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_5592_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_5593_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_5594_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_5595_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5596_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5597_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5598_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_pos
thf(fact_5599_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_5600_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_5601_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_5602_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_5603_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_5604_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_5605_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_5606_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_5607_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5608_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X3 ) ) ) ).

% floor_exists1
thf(fact_5609_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_5610_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W2 @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_5611_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_5612_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_5613_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5614_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5615_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5616_split__nat,axiom,
    ! [P: nat > $o,I2: int] :
      ( ( P @ ( nat2 @ I2 ) )
      = ( ! [N: nat] :
            ( ( I2
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P @ N ) )
        & ( ( ord_less_int @ I2 @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_5617_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_5618_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_5619_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_5620_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_5621_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_5622_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_5623_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_5624_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_5625_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_5626_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_5627_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_5628_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_5629_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_5630_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5631_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5632_ceiling__split,axiom,
    ! [P: int > $o,T5: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T5 ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T5 )
              & ( ord_less_eq_rat @ T5 @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_5633_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5634_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5635_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_5636_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_5637_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_5638_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_5639_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_5640_ceiling__divide__upper,axiom,
    ! [Q6: rat,P3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q6 )
     => ( ord_less_eq_rat @ P3 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P3 @ Q6 ) ) ) @ Q6 ) ) ) ).

% ceiling_divide_upper
thf(fact_5641_ceiling__divide__lower,axiom,
    ! [Q6: rat,P3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q6 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P3 @ Q6 ) ) ) @ one_one_rat ) @ Q6 ) @ P3 ) ) ).

% ceiling_divide_lower
thf(fact_5642_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_5643_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A22: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q7: int] :
              ( ( A12 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q7 @ L2 ) ) )
          | ? [R4: int,L2: int,K3: int,Q7: int] :
              ( ( A12 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q7 @ R4 ) )
              & ( ( sgn_sgn_int @ R4 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q7 @ L2 ) @ R4 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5644_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q5: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q5 @ A23 ) ) ) )
         => ~ ! [R2: int,Q5: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q5 @ R2 ) )
               => ( ( ( sgn_sgn_int @ R2 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A23 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A23 ) @ R2 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5645_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_5646_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5647_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5648_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_5649_sgn__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_less
thf(fact_5650_sgn__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_less
thf(fact_5651_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_5652_sgn__greater,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_greater
thf(fact_5653_sgn__greater,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_greater
thf(fact_5654_divide__sgn,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
      = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).

% divide_sgn
thf(fact_5655_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_5656_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_5657_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_5658_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
      = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_5659_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_5660_sgn__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_5661_sgn__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_5662_union__diff__assoc,axiom,
    ! [C3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat] :
      ( ( ( minus_4286766774447292334at_nat @ C3 @ B3 )
        = zero_z1048942125864253310at_nat )
     => ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A3 @ B3 ) @ C3 )
        = ( plus_p7104986032573967614at_nat @ A3 @ ( minus_4286766774447292334at_nat @ B3 @ C3 ) ) ) ) ).

% union_diff_assoc
thf(fact_5663_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).

% sgn_mult
thf(fact_5664_sgn__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_mult
thf(fact_5665_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_5666_Ints__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ B @ ring_11222124179247155820nteger )
       => ( member_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ring_11222124179247155820nteger ) ) ) ).

% Ints_mult
thf(fact_5667_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_mult
thf(fact_5668_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_mult
thf(fact_5669_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5670_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5671_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5672_abs__mult__sgn,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5673_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5674_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_5675_sgn__mult__abs,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5676_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5677_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_5678_mult__sgn__abs,axiom,
    ! [X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5679_mult__sgn__abs,axiom,
    ! [X: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5680_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5681_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_5682_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_5683_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_5684_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_1_pos
thf(fact_5685_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_1_pos
thf(fact_5686_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_5687_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_5688_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_5689_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_5690_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_5691_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X4: int] : ( if_int @ ( X4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_5692_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X4: code_integer] : ( if_Code_integer @ ( X4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X4 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_5693_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X4: rat] : ( if_rat @ ( X4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_5694_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A ) @ zero_z3403309356797280102nteger )
        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% Ints_odd_less_0
thf(fact_5695_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_5696_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_5697_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( X != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5698_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( X != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5699_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( X != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5700_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5701_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5702_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5703_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
       => ( ( X = Y )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5704_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( member_rat @ Y @ ring_1_Ints_rat )
       => ( ( X = Y )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5705_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( member_int @ Y @ ring_1_Ints_int )
       => ( ( X = Y )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5706_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_5707_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_5708_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q6: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q6 @ L ) @ R3 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q6 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_5709_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5710_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5711_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_5712_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs ) )
       => ( ( size_size_list_o @ ( slice_o @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_5713_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_5714_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% pochhammer_same
thf(fact_5715_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% pochhammer_same
thf(fact_5716_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% pochhammer_same
thf(fact_5717_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5718_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5719_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri3624122377584611663nteger @ N2 )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5720_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5721_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5722_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5723_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M3 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5724_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5725_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_5726_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate1_o @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_5727_cpmi,axiom,
    ! [D3: int,P: int > $o,P6: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B3 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X11: int] : ( P @ X11 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ B3 )
                        & ( P @ ( plus_plus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_5728_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5729_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5730_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5731_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5732_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5733_Icc__eq__Icc,axiom,
    ! [L: code_integer,H: code_integer,L3: code_integer,H3: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L @ H )
        = ( set_or189985376899183464nteger @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L @ H )
          & ~ ( ord_le3102999989581377725nteger @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5734_atLeastAtMost__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( ( member_o @ I2 @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I2 )
        & ( ord_less_eq_o @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5735_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I2 )
        & ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5736_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I2 )
        & ( ord_less_eq_set_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5737_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I2 )
        & ( ord_less_eq_rat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5738_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I2 )
        & ( ord_less_eq_num @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5739_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I2 )
        & ( ord_less_eq_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5740_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_eq_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5741_atLeastAtMost__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I2 @ ( set_or189985376899183464nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I2 )
        & ( ord_le3102999989581377725nteger @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5742_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5743_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5744_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5745_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5746_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5747_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5748_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A @ B ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5749_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5750_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5751_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5752_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5753_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5754_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5755_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5756_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_5757_atLeastatMost__empty,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( set_or7959216805967363635t_assn @ A @ B )
        = bot_bot_set_assn ) ) ).

% atLeastatMost_empty
thf(fact_5758_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_5759_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_5760_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_5761_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_5762_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_5763_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D2: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5764_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5765_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5766_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5767_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5768_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5769_slice__complete,axiom,
    ! [Xs: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_5770_slice__complete,axiom,
    ! [Xs: list_o] :
      ( ( slice_o @ zero_zero_nat @ ( size_size_list_o @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_5771_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_Suc
thf(fact_5772_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_5773_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri3624122377584611663nteger @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N2 ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% fact_Suc
thf(fact_5774_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_5775_mset__distrib,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A3 @ B3 )
        = ( plus_p7104986032573967614at_nat @ M6 @ N7 ) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A3
              = ( plus_p7104986032573967614at_nat @ Am @ An ) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B3
                  = ( plus_p7104986032573967614at_nat @ Bm @ Bn ) )
               => ( ( M6
                    = ( plus_p7104986032573967614at_nat @ Am @ Bm ) )
                 => ( N7
                   != ( plus_p7104986032573967614at_nat @ An @ Bn ) ) ) ) ) ) ).

% mset_distrib
thf(fact_5776_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_5777_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_5778_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_eq_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_5779_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_eq_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M ) @ ( set_or7049704709247886629st_num @ M @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_5780_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_eq_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_5781_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_eq_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_5782_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M )
     => ( ( ord_le3102999989581377725nteger @ M @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
          = ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_5783_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_5784_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_ge_zero
thf(fact_5785_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_5786_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_5787_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_5788_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% fact_not_neg
thf(fact_5789_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_5790_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_5791_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_5792_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_gt_zero
thf(fact_5793_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_5794_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_5795_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_5796_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_ge_1
thf(fact_5797_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_5798_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_5799_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_5800_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_mono
thf(fact_5801_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_5802_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_5803_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_5804_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_5805_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_5806_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_5807_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_5808_atLeastatMost__psubset__iff,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( ( ord_less_set_assn @ ( set_or7959216805967363635t_assn @ A @ B ) @ ( set_or7959216805967363635t_assn @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_assn @ A @ B )
          | ( ( ord_less_eq_assn @ C @ A )
            & ( ord_less_eq_assn @ B @ D2 )
            & ( ( ord_less_assn @ C @ A )
              | ( ord_less_assn @ B @ D2 ) ) ) )
        & ( ord_less_eq_assn @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5809_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D2: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D2 )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_set_int @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5810_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D2 )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D2 ) ) ) )
        & ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5811_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D2 )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D2 ) ) ) )
        & ( ord_less_eq_num @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5812_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D2 )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_int @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5813_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D2 )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_nat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5814_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D2 ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D2 )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5815_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_5816_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_5817_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_5818_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_5819_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_5820_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_5821_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N2 ) @ ( semiri2447717529341329178atural @ M ) )
        = zero_z2226904508553997617atural ) ) ).

% fact_mod
thf(fact_5822_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_5823_fact__le__power,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_5824_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_5825_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_5826_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_5827_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_5828_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_5829_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5830_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_5831_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R3 ) ) ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_5832_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_5833_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_5834_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A5 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_5835_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_5836_periodic__finite__ex,axiom,
    ! [D2: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ( ? [X11: int] : ( P @ X11 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P @ X4 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_5837_aset_I7_J,axiom,
    ! [D3: int,A3: set_int,T5: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T5 @ X7 )
           => ( ord_less_int @ T5 @ ( plus_plus_int @ X7 @ D3 ) ) ) ) ) ).

% aset(7)
thf(fact_5838_aset_I5_J,axiom,
    ! [D3: int,T5: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T5 @ A3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X7 @ T5 )
             => ( ord_less_int @ ( plus_plus_int @ X7 @ D3 ) @ T5 ) ) ) ) ) ).

% aset(5)
thf(fact_5839_aset_I4_J,axiom,
    ! [D3: int,T5: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T5 @ A3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 != T5 )
             => ( ( plus_plus_int @ X7 @ D3 )
               != T5 ) ) ) ) ) ).

% aset(4)
thf(fact_5840_aset_I3_J,axiom,
    ! [D3: int,T5: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T5 @ one_one_int ) @ A3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 = T5 )
             => ( ( plus_plus_int @ X7 @ D3 )
                = T5 ) ) ) ) ) ).

% aset(3)
thf(fact_5841_bset_I7_J,axiom,
    ! [D3: int,T5: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T5 @ B3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B3 )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T5 @ X7 )
             => ( ord_less_int @ T5 @ ( minus_minus_int @ X7 @ D3 ) ) ) ) ) ) ).

% bset(7)
thf(fact_5842_bset_I5_J,axiom,
    ! [D3: int,B3: set_int,T5: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B3 )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X7 @ T5 )
           => ( ord_less_int @ ( minus_minus_int @ X7 @ D3 ) @ T5 ) ) ) ) ).

% bset(5)
thf(fact_5843_bset_I4_J,axiom,
    ! [D3: int,T5: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T5 @ B3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B3 )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 != T5 )
             => ( ( minus_minus_int @ X7 @ D3 )
               != T5 ) ) ) ) ) ).

% bset(4)
thf(fact_5844_bset_I3_J,axiom,
    ! [D3: int,T5: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T5 @ one_one_int ) @ B3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B3 )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X7 = T5 )
             => ( ( minus_minus_int @ X7 @ D3 )
                = T5 ) ) ) ) ) ).

% bset(3)
thf(fact_5845_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_5846_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5847_aset_I8_J,axiom,
    ! [D3: int,A3: set_int,T5: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A3 )
                 => ( X7
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T5 @ X7 )
           => ( ord_less_eq_int @ T5 @ ( plus_plus_int @ X7 @ D3 ) ) ) ) ) ).

% aset(8)
thf(fact_5848_aset_I6_J,axiom,
    ! [D3: int,T5: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T5 @ one_one_int ) @ A3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A3 )
                   => ( X7
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X7 @ T5 )
             => ( ord_less_eq_int @ ( plus_plus_int @ X7 @ D3 ) @ T5 ) ) ) ) ) ).

% aset(6)
thf(fact_5849_bset_I8_J,axiom,
    ! [D3: int,T5: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T5 @ one_one_int ) @ B3 )
       => ! [X7: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B3 )
                   => ( X7
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T5 @ X7 )
             => ( ord_less_eq_int @ T5 @ ( minus_minus_int @ X7 @ D3 ) ) ) ) ) ) ).

% bset(8)
thf(fact_5850_bset_I6_J,axiom,
    ! [D3: int,B3: set_int,T5: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X7: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B3 )
                 => ( X7
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X7 @ T5 )
           => ( ord_less_eq_int @ ( minus_minus_int @ X7 @ D3 ) @ T5 ) ) ) ) ).

% bset(6)
thf(fact_5851_cppi,axiom,
    ! [D3: int,P: int > $o,P6: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A3 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X11: int] : ( P @ X11 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ A3 )
                        & ( P @ ( minus_minus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_5852_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I2: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_int @ ( slice_int @ From @ To @ Xs ) @ I2 )
            = ( nth_int @ Xs @ ( plus_plus_nat @ From @ I2 ) ) ) ) ) ) ).

% slice_nth
thf(fact_5853_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I2: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_nat @ ( slice_nat @ From @ To @ Xs ) @ I2 )
            = ( nth_nat @ Xs @ ( plus_plus_nat @ From @ I2 ) ) ) ) ) ) ).

% slice_nth
thf(fact_5854_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I2: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_o @ Xs ) )
       => ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_o @ ( slice_o @ From @ To @ Xs ) @ I2 )
            = ( nth_o @ Xs @ ( plus_plus_nat @ From @ I2 ) ) ) ) ) ) ).

% slice_nth
thf(fact_5855_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: rat,B: rat,M: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X4: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( divide_divide_rat @ A @ M ) @ C ) @ ( minus_minus_rat @ ( divide_divide_rat @ B @ M ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( divide_divide_rat @ B @ M ) @ C ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ M ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
thf(fact_5856_image__affinity__atLeastAtMost__div,axiom,
    ! [A: rat,B: rat,M: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X4: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( divide_divide_rat @ A @ M ) @ C ) @ ( plus_plus_rat @ ( divide_divide_rat @ B @ M ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X4 @ M ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( divide_divide_rat @ B @ M ) @ C ) @ ( plus_plus_rat @ ( divide_divide_rat @ A @ M ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
thf(fact_5857_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: rat,B: rat,M: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X4: rat] : ( minus_minus_rat @ ( times_times_rat @ M @ X4 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( minus_minus_rat @ ( times_times_rat @ M @ X4 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( times_times_rat @ M @ A ) @ C ) @ ( minus_minus_rat @ ( times_times_rat @ M @ B ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( minus_minus_rat @ ( times_times_rat @ M @ X4 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( times_times_rat @ M @ B ) @ C ) @ ( minus_minus_rat @ ( times_times_rat @ M @ A ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
thf(fact_5858_image__affinity__atLeastAtMost,axiom,
    ! [A: rat,B: rat,M: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X4: rat] : ( plus_plus_rat @ ( times_times_rat @ M @ X4 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( plus_plus_rat @ ( times_times_rat @ M @ X4 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( times_times_rat @ M @ A ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ M @ B ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( plus_plus_rat @ ( times_times_rat @ M @ X4 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( times_times_rat @ M @ B ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ M @ A ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
thf(fact_5859_image__mult__atLeastAtMost__if_H,axiom,
    ! [X: rat,Y: rat,C: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( times_times_rat @ X4 @ C )
                @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ X @ C ) @ ( times_times_rat @ Y @ C ) ) ) )
          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
           => ( ( image_rat_rat
                @ ^ [X4: rat] : ( times_times_rat @ X4 @ C )
                @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ Y @ C ) @ ( times_times_rat @ X @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_rat @ X @ Y )
       => ( ( image_rat_rat
            @ ^ [X4: rat] : ( times_times_rat @ X4 @ C )
            @ ( set_or633870826150836451st_rat @ X @ Y ) )
          = bot_bot_set_rat ) ) ) ).

% image_mult_atLeastAtMost_if'
thf(fact_5860_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_5861_image__eqI,axiom,
    ! [B: $o,F: $o > $o,X: $o,A3: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A3 )
       => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5862_image__eqI,axiom,
    ! [B: nat,F: $o > nat,X: $o,A3: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A3 )
       => ( member_nat @ B @ ( image_o_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5863_image__eqI,axiom,
    ! [B: int,F: $o > int,X: $o,A3: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A3 )
       => ( member_int @ B @ ( image_o_int @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5864_image__eqI,axiom,
    ! [B: $o,F: nat > $o,X: nat,A3: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A3 )
       => ( member_o @ B @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5865_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X: nat,A3: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A3 )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5866_image__eqI,axiom,
    ! [B: int,F: nat > int,X: nat,A3: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A3 )
       => ( member_int @ B @ ( image_nat_int @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5867_image__eqI,axiom,
    ! [B: $o,F: int > $o,X: int,A3: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int @ X @ A3 )
       => ( member_o @ B @ ( image_int_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5868_image__eqI,axiom,
    ! [B: nat,F: int > nat,X: int,A3: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int @ X @ A3 )
       => ( member_nat @ B @ ( image_int_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5869_image__eqI,axiom,
    ! [B: int,F: int > int,X: int,A3: set_int] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_int @ X @ A3 )
       => ( member_int @ B @ ( image_int_int @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5870_image__eqI,axiom,
    ! [B: set_nat,F: $o > set_nat,X: $o,A3: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A3 )
       => ( member_set_nat @ B @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_5871_image__ident,axiom,
    ! [Y7: set_nat] :
      ( ( image_nat_nat
        @ ^ [X4: nat] : X4
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_5872_image__ident,axiom,
    ! [Y7: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : X4
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_5873_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5874_image__empty,axiom,
    ! [F: $o > nat] :
      ( ( image_o_nat @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_5875_image__empty,axiom,
    ! [F: $o > int] :
      ( ( image_o_int @ F @ bot_bot_set_o )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_5876_image__empty,axiom,
    ! [F: nat > $o] :
      ( ( image_nat_o @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5877_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_5878_image__empty,axiom,
    ! [F: nat > int] :
      ( ( image_nat_int @ F @ bot_bot_set_nat )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_5879_image__empty,axiom,
    ! [F: int > $o] :
      ( ( image_int_o @ F @ bot_bot_set_int )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_5880_image__empty,axiom,
    ! [F: int > nat] :
      ( ( image_int_nat @ F @ bot_bot_set_int )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_5881_image__empty,axiom,
    ! [F: int > int] :
      ( ( image_int_int @ F @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_5882_image__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( image_nat_set_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_set_nat ) ).

% image_empty
thf(fact_5883_empty__is__image,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5884_empty__is__image,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5885_empty__is__image,axiom,
    ! [F: int > $o,A3: set_int] :
      ( ( bot_bot_set_o
        = ( image_int_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_5886_empty__is__image,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5887_empty__is__image,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5888_empty__is__image,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( bot_bot_set_nat
        = ( image_int_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_5889_empty__is__image,axiom,
    ! [F: $o > int,A3: set_o] :
      ( ( bot_bot_set_int
        = ( image_o_int @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_5890_empty__is__image,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( bot_bot_set_int
        = ( image_nat_int @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5891_empty__is__image,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( bot_bot_set_int
        = ( image_int_int @ F @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_5892_empty__is__image,axiom,
    ! [F: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( image_nat_set_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_5893_image__is__empty,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( ( image_o_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5894_image__is__empty,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( ( image_nat_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5895_image__is__empty,axiom,
    ! [F: int > $o,A3: set_int] :
      ( ( ( image_int_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_5896_image__is__empty,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( ( image_o_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5897_image__is__empty,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5898_image__is__empty,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( ( image_int_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_5899_image__is__empty,axiom,
    ! [F: $o > int,A3: set_o] :
      ( ( ( image_o_int @ F @ A3 )
        = bot_bot_set_int )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_5900_image__is__empty,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ( image_nat_int @ F @ A3 )
        = bot_bot_set_int )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5901_image__is__empty,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ( image_int_int @ F @ A3 )
        = bot_bot_set_int )
      = ( A3 = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_5902_image__is__empty,axiom,
    ! [F: nat > set_nat,A3: set_nat] :
      ( ( ( image_nat_set_nat @ F @ A3 )
        = bot_bot_set_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_5903_image__add__0,axiom,
    ! [S: set_Code_integer] :
      ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ) @ S )
      = S ) ).

% image_add_0
thf(fact_5904_image__add__0,axiom,
    ! [S: set_mu2057375006010111271at_nat] :
      ( ( image_2516384619306764229at_nat @ ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ) @ S )
      = S ) ).

% image_add_0
thf(fact_5905_image__add__0,axiom,
    ! [S: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S )
      = S ) ).

% image_add_0
thf(fact_5906_image__add__0,axiom,
    ! [S: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
      = S ) ).

% image_add_0
thf(fact_5907_image__add__0,axiom,
    ! [S: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S )
      = S ) ).

% image_add_0
thf(fact_5908_image__add__atLeastAtMost_H,axiom,
    ! [K: rat,I2: rat,J: rat] :
      ( ( image_rat_rat
        @ ^ [N: rat] : ( plus_plus_rat @ N @ K )
        @ ( set_or633870826150836451st_rat @ I2 @ J ) )
      = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_5909_image__add__atLeastAtMost_H,axiom,
    ! [K: int,I2: int,J: int] :
      ( ( image_int_int
        @ ^ [N: int] : ( plus_plus_int @ N @ K )
        @ ( set_or1266510415728281911st_int @ I2 @ J ) )
      = ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_5910_image__add__atLeastAtMost_H,axiom,
    ! [K: nat,I2: nat,J: nat] :
      ( ( image_nat_nat
        @ ^ [N: nat] : ( plus_plus_nat @ N @ K )
        @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_5911_image__add__atLeastAtMost_H,axiom,
    ! [K: code_integer,I2: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [N: code_integer] : ( plus_p5714425477246183910nteger @ N @ K )
        @ ( set_or189985376899183464nteger @ I2 @ J ) )
      = ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_5912_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( image_rat_rat
        @ ^ [T3: rat] : ( minus_minus_rat @ T3 @ D2 )
        @ ( set_or633870826150836451st_rat @ A @ B ) )
      = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ A @ D2 ) @ ( minus_minus_rat @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_5913_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( image_int_int
        @ ^ [T3: int] : ( minus_minus_int @ T3 @ D2 )
        @ ( set_or1266510415728281911st_int @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( minus_minus_int @ A @ D2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_5914_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: code_integer,A: code_integer,B: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [T3: code_integer] : ( minus_8373710615458151222nteger @ T3 @ D2 )
        @ ( set_or189985376899183464nteger @ A @ B ) )
      = ( set_or189985376899183464nteger @ ( minus_8373710615458151222nteger @ A @ D2 ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_5915_if__image__distrib,axiom,
    ! [P: int > $o,F: int > int,G: int > int,S: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( if_int @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_int @ ( image_int_int @ F @ ( inf_inf_set_int @ S @ ( collect_int @ P ) ) )
        @ ( image_int_int @ G
          @ ( inf_inf_set_int @ S
            @ ( collect_int
              @ ^ [X4: int] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5916_if__image__distrib,axiom,
    ! [P: nat > $o,F: nat > int,G: nat > int,S: set_nat] :
      ( ( image_nat_int
        @ ^ [X4: nat] : ( if_int @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_int @ ( image_nat_int @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_nat_int @ G
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5917_if__image__distrib,axiom,
    ! [P: int > $o,F: int > nat,G: int > nat,S: set_int] :
      ( ( image_int_nat
        @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_int_nat @ F @ ( inf_inf_set_int @ S @ ( collect_int @ P ) ) )
        @ ( image_int_nat @ G
          @ ( inf_inf_set_int @ S
            @ ( collect_int
              @ ^ [X4: int] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5918_if__image__distrib,axiom,
    ! [P: nat > $o,F: nat > nat,G: nat > nat,S: set_nat] :
      ( ( image_nat_nat
        @ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_nat_nat @ G
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5919_if__image__distrib,axiom,
    ! [P: nat > $o,F: nat > set_nat,G: nat > set_nat,S: set_nat] :
      ( ( image_nat_set_nat
        @ ^ [X4: nat] : ( if_set_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_nat_set_nat @ G
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5920_if__image__distrib,axiom,
    ! [P: list_nat > $o,F: list_nat > nat,G: list_nat > nat,S: set_list_nat] :
      ( ( image_list_nat_nat
        @ ^ [X4: list_nat] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_list_nat_nat @ F @ ( inf_inf_set_list_nat @ S @ ( collect_list_nat @ P ) ) )
        @ ( image_list_nat_nat @ G
          @ ( inf_inf_set_list_nat @ S
            @ ( collect_list_nat
              @ ^ [X4: list_nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5921_if__image__distrib,axiom,
    ! [P: set_nat > $o,F: set_nat > nat,G: set_nat > nat,S: set_set_nat] :
      ( ( image_set_nat_nat
        @ ^ [X4: set_nat] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ ( inf_inf_set_set_nat @ S @ ( collect_set_nat @ P ) ) )
        @ ( image_set_nat_nat @ G
          @ ( inf_inf_set_set_nat @ S
            @ ( collect_set_nat
              @ ^ [X4: set_nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5922_if__image__distrib,axiom,
    ! [P: product_prod_int_int > $o,F: product_prod_int_int > nat,G: product_prod_int_int > nat,S: set_Pr958786334691620121nt_int] :
      ( ( image_5044651549707136836nt_nat
        @ ^ [X4: product_prod_int_int] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_5044651549707136836nt_nat @ F @ ( inf_in2269163501485487111nt_int @ S @ ( collec213857154873943460nt_int @ P ) ) )
        @ ( image_5044651549707136836nt_nat @ G
          @ ( inf_in2269163501485487111nt_int @ S
            @ ( collec213857154873943460nt_int
              @ ^ [X4: product_prod_int_int] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5923_if__image__distrib,axiom,
    ! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat,S: set_Pr1261947904930325089at_nat] :
      ( ( image_2486076414777270412at_nat
        @ ^ [X4: product_prod_nat_nat] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_2486076414777270412at_nat @ F @ ( inf_in2572325071724192079at_nat @ S @ ( collec3392354462482085612at_nat @ P ) ) )
        @ ( image_2486076414777270412at_nat @ G
          @ ( inf_in2572325071724192079at_nat @ S
            @ ( collec3392354462482085612at_nat
              @ ^ [X4: product_prod_nat_nat] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5924_if__image__distrib,axiom,
    ! [P: int > $o,F: int > produc859450856879609959at_nat,G: int > produc859450856879609959at_nat,S: set_int] :
      ( ( image_6031046145702598900at_nat
        @ ^ [X4: int] : ( if_Pro4507677147265585453at_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
        @ S )
      = ( sup_su718114333110466843at_nat @ ( image_6031046145702598900at_nat @ F @ ( inf_inf_set_int @ S @ ( collect_int @ P ) ) )
        @ ( image_6031046145702598900at_nat @ G
          @ ( inf_inf_set_int @ S
            @ ( collect_int
              @ ^ [X4: int] :
                  ~ ( P @ X4 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_5925_image__mult__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D2 )
     => ( ( image_rat_rat @ ( times_times_rat @ D2 ) @ ( set_or633870826150836451st_rat @ A @ B ) )
        = ( set_or633870826150836451st_rat @ ( times_times_rat @ D2 @ A ) @ ( times_times_rat @ D2 @ B ) ) ) ) ).

% image_mult_atLeastAtMost
thf(fact_5926_image__divide__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D2 )
     => ( ( image_rat_rat
          @ ^ [C4: rat] : ( divide_divide_rat @ C4 @ D2 )
          @ ( set_or633870826150836451st_rat @ A @ B ) )
        = ( set_or633870826150836451st_rat @ ( divide_divide_rat @ A @ D2 ) @ ( divide_divide_rat @ B @ D2 ) ) ) ) ).

% image_divide_atLeastAtMost
thf(fact_5927_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: $o,F: $o > $o] :
      ( ( member_o @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5928_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: nat,F: $o > nat] :
      ( ( member_o @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_o_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5929_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: int,F: $o > int] :
      ( ( member_o @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_int @ B @ ( image_o_int @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5930_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: $o,F: nat > $o] :
      ( ( member_nat @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5931_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5932_rev__image__eqI,axiom,
    ! [X: nat,A3: set_nat,B: int,F: nat > int] :
      ( ( member_nat @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_int @ B @ ( image_nat_int @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5933_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: $o,F: int > $o] :
      ( ( member_int @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_int_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5934_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: nat,F: int > nat] :
      ( ( member_int @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_int_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5935_rev__image__eqI,axiom,
    ! [X: int,A3: set_int,B: int,F: int > int] :
      ( ( member_int @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_int @ B @ ( image_int_int @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5936_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B: set_nat,F: $o > set_nat] :
      ( ( member_o @ X @ A3 )
     => ( ( B
          = ( F @ X ) )
       => ( member_set_nat @ B @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_5937_ball__imageD,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ A3 )
         => ( P @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_5938_ball__imageD,axiom,
    ! [F: nat > nat,A3: set_nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ A3 )
         => ( P @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_5939_ball__imageD,axiom,
    ! [F: nat > int,A3: set_nat,P: int > $o] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( image_nat_int @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ A3 )
         => ( P @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_5940_ball__imageD,axiom,
    ! [F: int > nat,A3: set_int,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_int_nat @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X7: int] :
          ( ( member_int @ X7 @ A3 )
         => ( P @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_5941_ball__imageD,axiom,
    ! [F: int > int,A3: set_int,P: int > $o] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( image_int_int @ F @ A3 ) )
         => ( P @ X3 ) )
     => ! [X7: int] :
          ( ( member_int @ X7 @ A3 )
         => ( P @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_5942_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N7 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_set_nat @ F @ M6 )
          = ( image_nat_set_nat @ G @ N7 ) ) ) ) ).

% image_cong
thf(fact_5943_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N7 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_nat @ F @ M6 )
          = ( image_nat_nat @ G @ N7 ) ) ) ) ).

% image_cong
thf(fact_5944_image__cong,axiom,
    ! [M6: set_nat,N7: set_nat,F: nat > int,G: nat > int] :
      ( ( M6 = N7 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N7 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_int @ F @ M6 )
          = ( image_nat_int @ G @ N7 ) ) ) ) ).

% image_cong
thf(fact_5945_image__cong,axiom,
    ! [M6: set_int,N7: set_int,F: int > nat,G: int > nat] :
      ( ( M6 = N7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ N7 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_int_nat @ F @ M6 )
          = ( image_int_nat @ G @ N7 ) ) ) ) ).

% image_cong
thf(fact_5946_image__cong,axiom,
    ! [M6: set_int,N7: set_int,F: int > int,G: int > int] :
      ( ( M6 = N7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ N7 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_int_int @ F @ M6 )
          = ( image_int_int @ G @ N7 ) ) ) ) ).

% image_cong
thf(fact_5947_bex__imageD,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_nat > $o] :
      ( ? [X7: set_nat] :
          ( ( member_set_nat @ X7 @ ( image_nat_set_nat @ F @ A3 ) )
          & ( P @ X7 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_5948_bex__imageD,axiom,
    ! [F: nat > nat,A3: set_nat,P: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( image_nat_nat @ F @ A3 ) )
          & ( P @ X7 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_5949_bex__imageD,axiom,
    ! [F: nat > int,A3: set_nat,P: int > $o] :
      ( ? [X7: int] :
          ( ( member_int @ X7 @ ( image_nat_int @ F @ A3 ) )
          & ( P @ X7 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_5950_bex__imageD,axiom,
    ! [F: int > nat,A3: set_int,P: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( image_int_nat @ F @ A3 ) )
          & ( P @ X7 ) )
     => ? [X3: int] :
          ( ( member_int @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_5951_bex__imageD,axiom,
    ! [F: int > int,A3: set_int,P: int > $o] :
      ( ? [X7: int] :
          ( ( member_int @ X7 @ ( image_int_int @ F @ A3 ) )
          & ( P @ X7 ) )
     => ? [X3: int] :
          ( ( member_int @ X3 @ A3 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_5952_image__iff,axiom,
    ! [Z: set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A3 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_5953_image__iff,axiom,
    ! [Z: nat,F: nat > nat,A3: set_nat] :
      ( ( member_nat @ Z @ ( image_nat_nat @ F @ A3 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_5954_image__iff,axiom,
    ! [Z: nat,F: int > nat,A3: set_int] :
      ( ( member_nat @ Z @ ( image_int_nat @ F @ A3 ) )
      = ( ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_5955_image__iff,axiom,
    ! [Z: int,F: nat > int,A3: set_nat] :
      ( ( member_int @ Z @ ( image_nat_int @ F @ A3 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_5956_image__iff,axiom,
    ! [Z: int,F: int > int,A3: set_int] :
      ( ( member_int @ Z @ ( image_int_int @ F @ A3 ) )
      = ( ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_5957_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > $o] :
      ( ( member_o @ X @ A3 )
     => ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_5958_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > nat] :
      ( ( member_o @ X @ A3 )
     => ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_5959_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > int] :
      ( ( member_o @ X @ A3 )
     => ( member_int @ ( F @ X ) @ ( image_o_int @ F @ A3 ) ) ) ).

% imageI
thf(fact_5960_imageI,axiom,
    ! [X: nat,A3: set_nat,F: nat > $o] :
      ( ( member_nat @ X @ A3 )
     => ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_5961_imageI,axiom,
    ! [X: nat,A3: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A3 )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_5962_imageI,axiom,
    ! [X: nat,A3: set_nat,F: nat > int] :
      ( ( member_nat @ X @ A3 )
     => ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A3 ) ) ) ).

% imageI
thf(fact_5963_imageI,axiom,
    ! [X: int,A3: set_int,F: int > $o] :
      ( ( member_int @ X @ A3 )
     => ( member_o @ ( F @ X ) @ ( image_int_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_5964_imageI,axiom,
    ! [X: int,A3: set_int,F: int > nat] :
      ( ( member_int @ X @ A3 )
     => ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_5965_imageI,axiom,
    ! [X: int,A3: set_int,F: int > int] :
      ( ( member_int @ X @ A3 )
     => ( member_int @ ( F @ X ) @ ( image_int_int @ F @ A3 ) ) ) ).

% imageI
thf(fact_5966_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > set_nat] :
      ( ( member_o @ X @ A3 )
     => ( member_set_nat @ ( F @ X ) @ ( image_o_set_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_5967_imageE,axiom,
    ! [B: $o,F: $o > $o,A3: set_o] :
      ( ( member_o @ B @ ( image_o_o @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5968_imageE,axiom,
    ! [B: $o,F: nat > $o,A3: set_nat] :
      ( ( member_o @ B @ ( image_nat_o @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5969_imageE,axiom,
    ! [B: $o,F: int > $o,A3: set_int] :
      ( ( member_o @ B @ ( image_int_o @ F @ A3 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5970_imageE,axiom,
    ! [B: nat,F: $o > nat,A3: set_o] :
      ( ( member_nat @ B @ ( image_o_nat @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5971_imageE,axiom,
    ! [B: nat,F: nat > nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5972_imageE,axiom,
    ! [B: nat,F: int > nat,A3: set_int] :
      ( ( member_nat @ B @ ( image_int_nat @ F @ A3 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5973_imageE,axiom,
    ! [B: int,F: $o > int,A3: set_o] :
      ( ( member_int @ B @ ( image_o_int @ F @ A3 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5974_imageE,axiom,
    ! [B: int,F: nat > int,A3: set_nat] :
      ( ( member_int @ B @ ( image_nat_int @ F @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5975_imageE,axiom,
    ! [B: int,F: int > int,A3: set_int] :
      ( ( member_int @ B @ ( image_int_int @ F @ A3 ) )
     => ~ ! [X3: int] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_int @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5976_imageE,axiom,
    ! [B: $o,F: set_nat > $o,A3: set_set_nat] :
      ( ( member_o @ B @ ( image_set_nat_o @ F @ A3 ) )
     => ~ ! [X3: set_nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_set_nat @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_5977_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5978_image__image,axiom,
    ! [F: nat > nat,G: int > nat,A3: set_int] :
      ( ( image_nat_nat @ F @ ( image_int_nat @ G @ A3 ) )
      = ( image_int_nat
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5979_image__image,axiom,
    ! [F: nat > int,G: nat > nat,A3: set_nat] :
      ( ( image_nat_int @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5980_image__image,axiom,
    ! [F: nat > int,G: int > nat,A3: set_int] :
      ( ( image_nat_int @ F @ ( image_int_nat @ G @ A3 ) )
      = ( image_int_int
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5981_image__image,axiom,
    ! [F: int > nat,G: nat > int,A3: set_nat] :
      ( ( image_int_nat @ F @ ( image_nat_int @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5982_image__image,axiom,
    ! [F: int > nat,G: int > int,A3: set_int] :
      ( ( image_int_nat @ F @ ( image_int_int @ G @ A3 ) )
      = ( image_int_nat
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5983_image__image,axiom,
    ! [F: int > int,G: nat > int,A3: set_nat] :
      ( ( image_int_int @ F @ ( image_nat_int @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5984_image__image,axiom,
    ! [F: int > int,G: int > int,A3: set_int] :
      ( ( image_int_int @ F @ ( image_int_int @ G @ A3 ) )
      = ( image_int_int
        @ ^ [X4: int] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5985_image__image,axiom,
    ! [F: set_nat > nat,G: nat > set_nat,A3: set_nat] :
      ( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5986_image__image,axiom,
    ! [F: set_nat > int,G: nat > set_nat,A3: set_nat] :
      ( ( image_set_nat_int @ F @ ( image_nat_set_nat @ G @ A3 ) )
      = ( image_nat_int
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A3 ) ) ).

% image_image
thf(fact_5987_Compr__image__eq,axiom,
    ! [F: $o > $o,A3: set_o,P: $o > $o] :
      ( ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ ( image_o_o @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_o_o @ F
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5988_Compr__image__eq,axiom,
    ! [F: nat > $o,A3: set_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ ( image_nat_o @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_o @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5989_Compr__image__eq,axiom,
    ! [F: int > $o,A3: set_int,P: $o > $o] :
      ( ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ ( image_int_o @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_int_o @ F
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5990_Compr__image__eq,axiom,
    ! [F: $o > nat,A3: set_o,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_o_nat @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_o_nat @ F
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5991_Compr__image__eq,axiom,
    ! [F: nat > nat,A3: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5992_Compr__image__eq,axiom,
    ! [F: int > nat,A3: set_int,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_int_nat @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_int_nat @ F
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5993_Compr__image__eq,axiom,
    ! [F: $o > int,A3: set_o,P: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ ( image_o_int @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_o_int @ F
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5994_Compr__image__eq,axiom,
    ! [F: nat > int,A3: set_nat,P: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ ( image_nat_int @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_int @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5995_Compr__image__eq,axiom,
    ! [F: int > int,A3: set_int,P: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ ( image_int_int @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_int_int @ F
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5996_Compr__image__eq,axiom,
    ! [F: list_nat > $o,A3: set_list_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ ( image_list_nat_o @ F @ A3 ) )
            & ( P @ X4 ) ) )
      = ( image_list_nat_o @ F
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A3 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_5997_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_5998_subset__image__iff,axiom,
    ! [B3: set_set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B3
              = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_5999_subset__image__iff,axiom,
    ! [B3: set_nat,F: nat > nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B3
              = ( image_nat_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_6000_subset__image__iff,axiom,
    ! [B3: set_nat,F: int > nat,A3: set_int] :
      ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A3 ) )
      = ( ? [AA: set_int] :
            ( ( ord_less_eq_set_int @ AA @ A3 )
            & ( B3
              = ( image_int_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_6001_subset__image__iff,axiom,
    ! [B3: set_int,F: nat > int,A3: set_nat] :
      ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A3 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A3 )
            & ( B3
              = ( image_nat_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_6002_subset__image__iff,axiom,
    ! [B3: set_int,F: int > int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A3 ) )
      = ( ? [AA: set_int] :
            ( ( ord_less_eq_set_int @ AA @ A3 )
            & ( B3
              = ( image_int_int @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_6003_image__subset__iff,axiom,
    ! [F: nat > set_nat,A3: set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ B3 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( member_set_nat @ ( F @ X4 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_6004_image__subset__iff,axiom,
    ! [F: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( member_nat @ ( F @ X4 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_6005_image__subset__iff,axiom,
    ! [F: int > nat,A3: set_int,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A3 ) @ B3 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( member_nat @ ( F @ X4 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_6006_image__subset__iff,axiom,
    ! [F: nat > int,A3: set_nat,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B3 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( member_int @ ( F @ X4 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_6007_image__subset__iff,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ B3 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( member_int @ ( F @ X4 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_6008_subset__imageE,axiom,
    ! [B3: set_set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
           => ( B3
             != ( image_nat_set_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_6009_subset__imageE,axiom,
    ! [B3: set_nat,F: nat > nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
           => ( B3
             != ( image_nat_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_6010_subset__imageE,axiom,
    ! [B3: set_nat,F: int > nat,A3: set_int] :
      ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A3 ) )
     => ~ ! [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A3 )
           => ( B3
             != ( image_int_nat @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_6011_subset__imageE,axiom,
    ! [B3: set_int,F: nat > int,A3: set_nat] :
      ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A3 ) )
     => ~ ! [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
           => ( B3
             != ( image_nat_int @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_6012_subset__imageE,axiom,
    ! [B3: set_int,F: int > int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A3 ) )
     => ~ ! [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A3 )
           => ( B3
             != ( image_int_int @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_6013_image__subsetI,axiom,
    ! [A3: set_o,F: $o > $o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_o_o @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6014_image__subsetI,axiom,
    ! [A3: set_o,F: $o > nat,B3: set_nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6015_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > $o,B3: set_o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_nat_o @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6016_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6017_image__subsetI,axiom,
    ! [A3: set_int,F: int > $o,B3: set_o] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_int_o @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6018_image__subsetI,axiom,
    ! [A3: set_int,F: int > nat,B3: set_nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6019_image__subsetI,axiom,
    ! [A3: set_o,F: $o > int,B3: set_int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( member_int @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_o_int @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6020_image__subsetI,axiom,
    ! [A3: set_nat,F: nat > int,B3: set_int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( member_int @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6021_image__subsetI,axiom,
    ! [A3: set_int,F: int > int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( member_int @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6022_image__subsetI,axiom,
    ! [A3: set_o,F: $o > set_nat,B3: set_set_nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( member_set_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A3 ) @ B3 ) ) ).

% image_subsetI
thf(fact_6023_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).

% image_mono
thf(fact_6024_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).

% image_mono
thf(fact_6025_image__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > int] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B3 ) ) ) ).

% image_mono
thf(fact_6026_image__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > nat] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A3 ) @ ( image_int_nat @ F @ B3 ) ) ) ).

% image_mono
thf(fact_6027_image__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B3 ) ) ) ).

% image_mono
thf(fact_6028_image__Un,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] :
      ( ( image_int_int @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
      = ( sup_sup_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6029_image__Un,axiom,
    ! [F: int > nat,A3: set_int,B3: set_int] :
      ( ( image_int_nat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( image_int_nat @ F @ A3 ) @ ( image_int_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6030_image__Un,axiom,
    ! [F: nat > int,A3: set_nat,B3: set_nat] :
      ( ( image_nat_int @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6031_image__Un,axiom,
    ! [F: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6032_image__Un,axiom,
    ! [F: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6033_image__Un,axiom,
    ! [F: produc859450856879609959at_nat > nat,A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( image_6160980552072806642at_nat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( image_6160980552072806642at_nat @ F @ A3 ) @ ( image_6160980552072806642at_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6034_image__Un,axiom,
    ! [F: nat > produc859450856879609959at_nat,A3: set_nat,B3: set_nat] :
      ( ( image_3276603626957510736at_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_su718114333110466843at_nat @ ( image_3276603626957510736at_nat @ F @ A3 ) @ ( image_3276603626957510736at_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6035_image__Un,axiom,
    ! [F: produc4166570645942440679at_nat > nat,A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
      ( ( image_6009382790362589554at_nat @ F @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( image_6009382790362589554at_nat @ F @ A3 ) @ ( image_6009382790362589554at_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6036_image__Un,axiom,
    ! [F: produc3843707927480180839at_nat > nat,A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( image_3198525501578754290at_nat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( image_3198525501578754290at_nat @ F @ A3 ) @ ( image_3198525501578754290at_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6037_image__Un,axiom,
    ! [F: nat > produc4166570645942440679at_nat,A3: set_nat,B3: set_nat] :
      ( ( image_5319605298250079696at_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_su3035147773818789531at_nat @ ( image_5319605298250079696at_nat @ F @ A3 ) @ ( image_5319605298250079696at_nat @ F @ B3 ) ) ) ).

% image_Un
thf(fact_6038_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > $o,B3: set_o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6039_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6040_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > $o,B3: set_o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_int_o @ F @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6041_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > nat,B3: set_nat] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6042_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > int,B3: set_int] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_int @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6043_image__Collect__subsetI,axiom,
    ! [P: int > $o,F: int > int,B3: set_int] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( member_int @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_int @ ( image_int_int @ F @ ( collect_int @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6044_image__Collect__subsetI,axiom,
    ! [P: list_nat > $o,F: list_nat > $o,B3: set_o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_list_nat_o @ F @ ( collect_list_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6045_image__Collect__subsetI,axiom,
    ! [P: list_nat > $o,F: list_nat > nat,B3: set_nat] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F @ ( collect_list_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6046_image__Collect__subsetI,axiom,
    ! [P: set_nat > $o,F: set_nat > $o,B3: set_o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_set_nat_o @ F @ ( collect_set_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6047_image__Collect__subsetI,axiom,
    ! [P: set_nat > $o,F: set_nat > nat,B3: set_nat] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ ( collect_set_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_6048_obtain__list__from__elements,axiom,
    ! [N2: nat,P: int > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: int] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_int] :
            ( ( ( size_size_list_int @ L4 )
              = N2 )
           => ~ ! [I: nat] :
                  ( ( ord_less_nat @ I @ N2 )
                 => ( P @ ( nth_int @ L4 @ I ) @ I ) ) ) ) ).

% obtain_list_from_elements
thf(fact_6049_obtain__list__from__elements,axiom,
    ! [N2: nat,P: nat > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: nat] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_nat] :
            ( ( ( size_size_list_nat @ L4 )
              = N2 )
           => ~ ! [I: nat] :
                  ( ( ord_less_nat @ I @ N2 )
                 => ( P @ ( nth_nat @ L4 @ I ) @ I ) ) ) ) ).

% obtain_list_from_elements
thf(fact_6050_obtain__list__from__elements,axiom,
    ! [N2: nat,P: $o > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: $o] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_o] :
            ( ( ( size_size_list_o @ L4 )
              = N2 )
           => ~ ! [I: nat] :
                  ( ( ord_less_nat @ I @ N2 )
                 => ( P @ ( nth_o @ L4 @ I ) @ I ) ) ) ) ).

% obtain_list_from_elements
thf(fact_6051_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z3: list_int] : Y5 = Z3 )
    = ( ^ [Xs3: list_int,Ys2: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys2 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I4 )
                = ( nth_int @ Ys2 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_6052_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z3: list_nat] : Y5 = Z3 )
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I4 )
                = ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_6053_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z3: list_o] : Y5 = Z3 )
    = ( ^ [Xs3: list_o,Ys2: list_o] :
          ( ( ( size_size_list_o @ Xs3 )
            = ( size_size_list_o @ Ys2 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs3 ) )
             => ( ( nth_o @ Xs3 @ I4 )
                = ( nth_o @ Ys2 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_6054_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X11: int] : ( P @ I4 @ X11 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_6055_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X11: nat] : ( P @ I4 @ X11 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_6056_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X11: $o] : ( P @ I4 @ X11 ) ) )
      = ( ? [Xs3: list_o] :
            ( ( ( size_size_list_o @ Xs3 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_o @ Xs3 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_6057_nth__equalityI,axiom,
    ! [Xs: list_int,Ys3: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys3 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I3 )
              = ( nth_int @ Ys3 @ I3 ) ) )
       => ( Xs = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_6058_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys3 @ I3 ) ) )
       => ( Xs = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_6059_nth__equalityI,axiom,
    ! [Xs: list_o,Ys3: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys3 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I3 )
              = ( nth_o @ Ys3 @ I3 ) ) )
       => ( Xs = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_6060_image__Int__subset,axiom,
    ! [F: int > nat,A3: set_int,B3: set_int] : ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( inf_inf_set_nat @ ( image_int_nat @ F @ A3 ) @ ( image_int_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6061_image__Int__subset,axiom,
    ! [F: nat > set_nat,A3: set_nat,B3: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6062_image__Int__subset,axiom,
    ! [F: nat > nat,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6063_image__Int__subset,axiom,
    ! [F: nat > product_prod_nat_nat,A3: set_nat,B3: set_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( inf_in2572325071724192079at_nat @ ( image_5846123807819985514at_nat @ F @ A3 ) @ ( image_5846123807819985514at_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6064_image__Int__subset,axiom,
    ! [F: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) @ ( inf_inf_set_nat @ ( image_2486076414777270412at_nat @ F @ A3 ) @ ( image_2486076414777270412at_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6065_image__Int__subset,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) @ ( inf_in2572325071724192079at_nat @ ( image_5168914502847457605at_nat @ F @ A3 ) @ ( image_5168914502847457605at_nat @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6066_image__Int__subset,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( inf_inf_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6067_image__Int__subset,axiom,
    ! [F: nat > int,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( inf_inf_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6068_image__Int__subset,axiom,
    ! [F: product_prod_nat_nat > int,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_int @ ( image_2483585944268220136at_int @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) @ ( inf_inf_set_int @ ( image_2483585944268220136at_int @ F @ A3 ) @ ( image_2483585944268220136at_int @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_6069_image__diff__subset,axiom,
    ! [F: nat > set_nat,A3: set_nat,B3: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_6070_image__diff__subset,axiom,
    ! [F: int > nat,A3: set_int,B3: set_int] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_int_nat @ F @ A3 ) @ ( image_int_nat @ F @ B3 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_6071_image__diff__subset,axiom,
    ! [F: nat > nat,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_6072_image__diff__subset,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B3 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_6073_image__diff__subset,axiom,
    ! [F: nat > int,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B3 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_6074_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( P @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_6075_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
           => ( P @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_6076_None__notin__image__Some,axiom,
    ! [A3: set_Pr7098220151150636591it_nat] :
      ~ ( member1310632352865511136it_nat @ none_P2651198173097904984it_nat @ ( image_2003495143432273353it_nat @ some_P7913643980934408916it_nat @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6077_None__notin__image__Some,axiom,
    ! [A3: set_num] :
      ~ ( member_option_num @ none_num @ ( image_num_option_num @ some_num @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6078_None__notin__image__Some,axiom,
    ! [A3: set_Pr1354866905816374718it_nat] :
      ~ ( member5596548051065438575it_nat @ none_P9117596204409417319it_nat @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6079_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate1_int @ Xs ) @ N2 )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_6080_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_6081_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( rotate1_o @ Xs ) @ N2 )
        = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_6082_image__mult__atLeastAtMost__if,axiom,
    ! [C: rat,X: rat,Y: rat] :
      ( ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
          = ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ X ) @ ( times_times_rat @ C @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ Y ) @ ( times_times_rat @ C @ X ) ) ) )
          & ( ~ ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = bot_bot_set_rat ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
thf(fact_6083_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_6084_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_6085_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys3: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys3 ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6086_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6087_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_int_o @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6088_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys3: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys3 ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6089_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6090_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6091_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys3: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys3 ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6092_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6093_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys3 ) @ N2 )
        = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6094_product__nth,axiom,
    ! [N2: nat,Xs: list_a,Ys3: list_P131111800688179804it_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s341701280123345136it_nat @ Ys3 ) ) )
     => ( ( nth_Pr8687047199098054322it_nat @ ( produc6966004487978991557it_nat @ Xs @ Ys3 ) @ N2 )
        = ( produc9178034014595674355it_nat @ ( nth_a @ Xs @ ( divide_divide_nat @ N2 @ ( size_s341701280123345136it_nat @ Ys3 ) ) ) @ ( nth_Pr4561283279332054789it_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_s341701280123345136it_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_6095_translation__subtract__Compl,axiom,
    ! [A: rat,T5: set_rat] :
      ( ( image_rat_rat
        @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
        @ ( uminus2201863774496077783et_rat @ T5 ) )
      = ( uminus2201863774496077783et_rat
        @ ( image_rat_rat
          @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_Compl
thf(fact_6096_translation__subtract__Compl,axiom,
    ! [A: int,T5: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
        @ ( uminus1532241313380277803et_int @ T5 ) )
      = ( uminus1532241313380277803et_int
        @ ( image_int_int
          @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_Compl
thf(fact_6097_translation__subtract__diff,axiom,
    ! [A: rat,S2: set_rat,T5: set_rat] :
      ( ( image_rat_rat
        @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
        @ ( minus_minus_set_rat @ S2 @ T5 ) )
      = ( minus_minus_set_rat
        @ ( image_rat_rat
          @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
          @ S2 )
        @ ( image_rat_rat
          @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_diff
thf(fact_6098_translation__subtract__diff,axiom,
    ! [A: int,S2: set_int,T5: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
        @ ( minus_minus_set_int @ S2 @ T5 ) )
      = ( minus_minus_set_int
        @ ( image_int_int
          @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
          @ S2 )
        @ ( image_int_int
          @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_diff
thf(fact_6099_translation__subtract__Int,axiom,
    ! [A: rat,S2: set_rat,T5: set_rat] :
      ( ( image_rat_rat
        @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
        @ ( inf_inf_set_rat @ S2 @ T5 ) )
      = ( inf_inf_set_rat
        @ ( image_rat_rat
          @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
          @ S2 )
        @ ( image_rat_rat
          @ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_Int
thf(fact_6100_translation__subtract__Int,axiom,
    ! [A: int,S2: set_int,T5: set_int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
        @ ( inf_inf_set_int @ S2 @ T5 ) )
      = ( inf_inf_set_int
        @ ( image_int_int
          @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
          @ S2 )
        @ ( image_int_int
          @ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
          @ T5 ) ) ) ).

% translation_subtract_Int
thf(fact_6101_sum__gp,axiom,
    ! [N2: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6102_of__nat__sum,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_6103_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_6104_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups7501900531339628137nteger
        @ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_6105_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_6106_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_6107_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_6108_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6109_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6110_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6111_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6112_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6113_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A3: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6114_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A3: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6115_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6116_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6117_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6118_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ ( suc @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6119_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6120_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6121_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6122_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6123_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6124_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6125_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6126_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M )
          @ ( groups6857163185585827899at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6127_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6128_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6129_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6130_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6131_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6132_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I5: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I4: nat] : ( power_8256067586552552935nteger @ X @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6133_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6134_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6135_sum__atLeastAtMost__code,axiom,
    ! [F: nat > code_integer,A: nat,B: nat] :
      ( ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A5: nat] : ( plus_p5714425477246183910nteger @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_z3403309356797280102nteger ) ) ).

% sum_atLeastAtMost_code
thf(fact_6136_sum__atLeastAtMost__code,axiom,
    ! [F: nat > multis2468970476368604999at_nat,A: nat,B: nat] :
      ( ( groups6857163185585827899at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo4497565046347964853at_nat
        @ ^ [A5: nat] : ( plus_p7104986032573967614at_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_z1048942125864253310at_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6137_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6138_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_6139_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6140_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6141_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6142_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6143_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6144_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > code_integer] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups7501900531339628137nteger
            @ ^ [K3: nat] : ( minus_8373710615458151222nteger @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_8373710615458151222nteger @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups7501900531339628137nteger
            @ ^ [K3: nat] : ( minus_8373710615458151222nteger @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% sum_natinterval_diff
thf(fact_6145_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6146_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6147_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6148_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6149_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6150_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6151_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6152_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_6153_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6154_translation__Int,axiom,
    ! [A: rat,S2: set_rat,T5: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( inf_inf_set_rat @ S2 @ T5 ) )
      = ( inf_inf_set_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ S2 ) @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T5 ) ) ) ).

% translation_Int
thf(fact_6155_translation__Int,axiom,
    ! [A: int,S2: set_int,T5: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ A ) @ ( inf_inf_set_int @ S2 @ T5 ) )
      = ( inf_inf_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S2 ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T5 ) ) ) ).

% translation_Int
thf(fact_6156_sum__abs__ge__zero,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_6157_sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_6158_sum_Oempty,axiom,
    ! [G: $o > code_integer] :
      ( ( groups4406642042086082107nteger @ G @ bot_bot_set_o )
      = zero_z3403309356797280102nteger ) ).

% sum.empty
thf(fact_6159_sum_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6160_sum_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6161_sum_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6162_sum_Oempty,axiom,
    ! [G: nat > code_integer] :
      ( ( groups7501900531339628137nteger @ G @ bot_bot_set_nat )
      = zero_z3403309356797280102nteger ) ).

% sum.empty
thf(fact_6163_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6164_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6165_sum_Oempty,axiom,
    ! [G: int > code_integer] :
      ( ( groups7873554091576472773nteger @ G @ bot_bot_set_int )
      = zero_z3403309356797280102nteger ) ).

% sum.empty
thf(fact_6166_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6167_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6168_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: $o > code_integer,A: $o > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups4406642042086082107nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups4406642042086082107nteger
                  @ ^ [I4: $o] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6169_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6170_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6171_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: $o > rat,A: $o > rat,B: rat,Delta: rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups7872700643590313910_o_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups7872700643590313910_o_rat
                  @ ^ [I4: $o] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6172_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6173_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6174_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: $o > int,A: $o > int,B: int,Delta: int] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups8505340233167759370_o_int @ X @ I5 )
          = one_one_int )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups8505340233167759370_o_int
                  @ ^ [I4: $o] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6175_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups3539618377306564664at_int @ X @ I5 )
          = one_one_int )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6176_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > int,A: int > int,B: int,Delta: int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups4538972089207619220nt_int @ X @ I5 )
          = one_one_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups4538972089207619220nt_int
                  @ ^ [I4: int] : ( times_times_int @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6177_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,X: set_nat > code_integer,A: set_nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: set_nat] :
          ( ( member_set_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups9190459664516455967nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: set_nat] :
              ( ( member_set_nat @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups9190459664516455967nteger
                  @ ^ [I4: set_nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6178_abs__sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( abs_abs_int
        @ ( groups4538972089207619220nt_int
          @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
          @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
        @ A3 ) ) ).

% abs_sum_abs
thf(fact_6179_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu2: nat] : zero_zero_nat
        @ A3 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_6180_sum_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu2: int] : zero_zero_int
        @ A3 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_6181_int__sum,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% int_sum
thf(fact_6182_int__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% int_sum
thf(fact_6183_sum__subtractf__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups977919841031483927at_nat
          @ ^ [X4: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6184_sum__subtractf__nat,axiom,
    ! [A3: set_o,G: $o > nat,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6185_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( groups8294997508430121362at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6186_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6187_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6188_sum__SucD,axiom,
    ! [F: nat > nat,A3: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A3 )
        = ( suc @ N2 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_6189_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6190_sum_Oswap,axiom,
    ! [G: int > int > int,B3: set_int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( groups4538972089207619220nt_int @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups4538972089207619220nt_int
        @ ^ [J3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I4: int] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6191_sum__mono,axiom,
    ! [K5: set_o,F: $o > rat,G: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K5 ) @ ( groups7872700643590313910_o_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6192_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6193_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6194_sum__mono,axiom,
    ! [K5: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K5 ) @ ( groups8507830703676809646_o_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6195_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6196_sum__mono,axiom,
    ! [K5: set_o,F: $o > int,G: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K5 ) @ ( groups8505340233167759370_o_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6197_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6198_sum__mono,axiom,
    ! [K5: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6199_sum__mono,axiom,
    ! [K5: set_int,F: int > int,G: int > int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6200_sum__mono,axiom,
    ! [K5: set_set_nat,F: set_nat > rat,G: set_nat > rat] :
      ( ! [I3: set_nat] :
          ( ( member_set_nat @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups7659867448343625626at_rat @ F @ K5 ) @ ( groups7659867448343625626at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6201_sum_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ H @ A3 ) ) ) ).

% sum.distrib
thf(fact_6202_sum_Odistrib,axiom,
    ! [G: int > int,H: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A3 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A3 ) @ ( groups4538972089207619220nt_int @ H @ A3 ) ) ) ).

% sum.distrib
thf(fact_6203_sum__product,axiom,
    ! [F: nat > nat,A3: set_nat,G: nat > nat,B3: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B3 )
        @ A3 ) ) ).

% sum_product
thf(fact_6204_sum__product,axiom,
    ! [F: int > int,A3: set_int,G: int > int,B3: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I4: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B3 )
        @ A3 ) ) ).

% sum_product
thf(fact_6205_sum__distrib__right,axiom,
    ! [F: nat > nat,A3: set_nat,R3: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ R3 )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_6206_sum__distrib__right,axiom,
    ! [F: int > int,A3: set_int,R3: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ R3 )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R3 )
        @ A3 ) ) ).

% sum_distrib_right
thf(fact_6207_sum__distrib__left,axiom,
    ! [R3: nat,F: nat > nat,A3: set_nat] :
      ( ( times_times_nat @ R3 @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ R3 @ ( F @ N ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_6208_sum__distrib__left,axiom,
    ! [R3: int,F: int > int,A3: set_int] :
      ( ( times_times_int @ R3 @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ R3 @ ( F @ N ) )
        @ A3 ) ) ).

% sum_distrib_left
thf(fact_6209_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( minus_minus_int @ ( F @ X4 ) @ ( G @ X4 ) )
        @ A3 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ G @ A3 ) ) ) ).

% sum_subtractf
thf(fact_6210_sum__negf,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( uminus_uminus_int @ ( F @ X4 ) )
        @ A3 )
      = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) ) ) ).

% sum_negf
thf(fact_6211_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups4406642042086082107nteger @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6212_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6213_sum__nonneg,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6214_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6215_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6216_sum__nonneg,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6217_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6218_sum__nonneg,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6219_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6220_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6221_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A3 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_6222_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A3 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_6223_sum__nonpos,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A3 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_6224_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6225_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6226_sum__nonpos,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6227_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6228_sum__nonpos,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6229_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A3 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_6230_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_6231_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6232_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_6233_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N ) @ ( modulo_modulo_nat @ M3 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_6234_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_o,X: $o] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ L ) )
         => ( ( sorted_wrt_o @ ord_less_eq_o @ L )
           => ( ( ord_less_eq_o @ ( nth_o @ L @ I2 ) @ X )
             => ( ( ord_less_o @ X @ ( nth_o @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_o @ ( nth_o @ L @ K2 ) @ X )
                         => ~ ( ord_less_o @ X @ ( nth_o @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_6235_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_rat,X: rat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_rat @ L ) )
         => ( ( sorted_wrt_rat @ ord_less_eq_rat @ L )
           => ( ( ord_less_eq_rat @ ( nth_rat @ L @ I2 ) @ X )
             => ( ( ord_less_rat @ X @ ( nth_rat @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_rat @ ( nth_rat @ L @ K2 ) @ X )
                         => ~ ( ord_less_rat @ X @ ( nth_rat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_6236_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_num,X: num] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ 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 @ I2 ) @ X )
             => ( ( ord_less_num @ X @ ( nth_num @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_num @ ( nth_num @ L @ K2 ) @ X )
                         => ~ ( ord_less_num @ X @ ( nth_num @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_6237_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ 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 @ I2 ) @ X )
             => ( ( ord_less_nat @ X @ ( nth_nat @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_nat @ ( nth_nat @ L @ K2 ) @ X )
                         => ~ ( ord_less_nat @ X @ ( nth_nat @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_6238_sorted__in__between,axiom,
    ! [I2: nat,J: nat,L: list_int,X: int] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ 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 @ I2 ) @ X )
             => ( ( ord_less_int @ X @ ( nth_int @ L @ J ) )
               => ~ ! [K2: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K2 )
                     => ( ( ord_less_nat @ K2 @ J )
                       => ( ( ord_less_eq_int @ ( nth_int @ L @ K2 ) @ X )
                         => ~ ( ord_less_int @ X @ ( nth_int @ L @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_6239_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_6240_gbinomial__Suc,axiom,
    ! [A: code_natural,K: nat] :
      ( ( gbinom7368847122466276068atural @ A @ ( suc @ K ) )
      = ( divide5121882707175180666atural
        @ ( groups2279045934846249631atural
          @ ^ [I4: nat] : ( minus_7197305767214868737atural @ A @ ( semiri3763490453095760265atural @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2447717529341329178atural @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_6241_gbinomial__Suc,axiom,
    ! [A: code_integer,K: nat] :
      ( ( gbinom8545251970709558553nteger @ A @ ( suc @ K ) )
      = ( divide6298287555418463151nteger
        @ ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( minus_8373710615458151222nteger @ A @ ( semiri4939895301339042750nteger @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri3624122377584611663nteger @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_6242_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_6243_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_6244_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_6245_atMost__iff,axiom,
    ! [I2: $o,K: $o] :
      ( ( member_o @ I2 @ ( set_ord_atMost_o @ K ) )
      = ( ord_less_eq_o @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6246_atMost__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6247_atMost__iff,axiom,
    ! [I2: set_int,K: set_int] :
      ( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K ) )
      = ( ord_less_eq_set_int @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6248_atMost__iff,axiom,
    ! [I2: rat,K: rat] :
      ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6249_atMost__iff,axiom,
    ! [I2: num,K: num] :
      ( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6250_atMost__iff,axiom,
    ! [I2: nat,K: nat] :
      ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6251_atMost__iff,axiom,
    ! [I2: int,K: int] :
      ( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I2 @ K ) ) ).

% atMost_iff
thf(fact_6252_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu2: nat] : one_one_nat
        @ A3 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_6253_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu2: nat] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_6254_prod_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu2: int] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_6255_of__nat__prod,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_6256_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups3455450783089532116nteger
        @ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_6257_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_6258_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_6259_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X4: nat] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_6260_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_6261_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_6262_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_6263_prod_Oempty,axiom,
    ! [G: $o > code_integer] :
      ( ( groups7694694392188491536nteger @ G @ bot_bot_set_o )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_6264_prod_Oempty,axiom,
    ! [G: $o > assn] :
      ( ( groups5301882518646026715o_assn @ G @ bot_bot_set_o )
      = one_one_assn ) ).

% prod.empty
thf(fact_6265_prod_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups2869687844427037835_o_rat @ G @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty
thf(fact_6266_prod_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty
thf(fact_6267_prod_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty
thf(fact_6268_prod_Oempty,axiom,
    ! [G: nat > code_integer] :
      ( ( groups3455450783089532116nteger @ G @ bot_bot_set_nat )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_6269_prod_Oempty,axiom,
    ! [G: nat > assn] :
      ( ( groups6906906614972039071t_assn @ G @ bot_bot_set_nat )
      = one_one_assn ) ).

% prod.empty
thf(fact_6270_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_6271_prod_Oempty,axiom,
    ! [G: int > code_integer] :
      ( ( groups3827104343326376752nteger @ G @ bot_bot_set_int )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_6272_prod_Oempty,axiom,
    ! [G: int > assn] :
      ( ( groups7882442080178216443t_assn @ G @ bot_bot_set_int )
      = one_one_assn ) ).

% prod.empty
thf(fact_6273_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6274_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6275_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6276_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6277_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6278_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H ) @ ( set_or58775011639299419et_int @ H3 ) )
      = ( ~ ( ord_less_eq_set_int @ L @ H )
        | ( ord_less_eq_set_int @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6279_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H )
        | ( ord_less_eq_rat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6280_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H )
        | ( ord_less_eq_num @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6281_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H )
        | ( ord_less_eq_int @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6282_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H )
        | ( ord_less_eq_nat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6283_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,H: code_integer,H3: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ L @ H ) @ ( set_or9101266186257409494nteger @ H3 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ L @ H )
        | ( ord_le3102999989581377725nteger @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6284_prod_OatMost__Suc,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6285_prod_OatMost__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6286_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6287_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6288_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6289_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6290_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6291_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6292_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6293_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6294_prod_Oswap,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% prod.swap
thf(fact_6295_prod_Oswap,axiom,
    ! [G: nat > nat > int,B3: set_nat,A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( groups705719431365010083at_int @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% prod.swap
thf(fact_6296_prod_Oswap,axiom,
    ! [G: nat > int > int,B3: set_int,A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( groups1705073143266064639nt_int @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% prod.swap
thf(fact_6297_prod_Oswap,axiom,
    ! [G: int > nat > int,B3: set_nat,A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I4: int] : ( groups705719431365010083at_int @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups1705073143266064639nt_int
            @ ^ [I4: int] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% prod.swap
thf(fact_6298_prod_Oswap,axiom,
    ! [G: int > int > int,B3: set_int,A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I4: int] : ( groups1705073143266064639nt_int @ ( G @ I4 ) @ B3 )
        @ A3 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups1705073143266064639nt_int
            @ ^ [I4: int] : ( G @ I4 @ J3 )
            @ A3 )
        @ B3 ) ) ).

% prod.swap
thf(fact_6299_sorted__wrt__true,axiom,
    ! [Xs: list_nat] :
      ( sorted_wrt_nat
      @ ^ [Uu2: nat,Uv: nat] : $true
      @ Xs ) ).

% sorted_wrt_true
thf(fact_6300_sorted__wrt__true,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int
      @ ^ [Uu2: int,Uv: int] : $true
      @ Xs ) ).

% sorted_wrt_true
thf(fact_6301_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: $o] :
      ( bot_bot_set_o
     != ( set_ord_atMost_o @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6302_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6303_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6304_prod_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( times_times_nat @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A3 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A3 ) @ ( groups708209901874060359at_nat @ H @ A3 ) ) ) ).

% prod.distrib
thf(fact_6305_prod_Odistrib,axiom,
    ! [G: nat > int,H: nat > int,A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( times_times_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A3 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A3 ) @ ( groups705719431365010083at_int @ H @ A3 ) ) ) ).

% prod.distrib
thf(fact_6306_prod_Odistrib,axiom,
    ! [G: int > int,H: int > int,A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( times_times_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A3 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A3 ) @ ( groups1705073143266064639nt_int @ H @ A3 ) ) ) ).

% prod.distrib
thf(fact_6307_prod__power__distrib,axiom,
    ! [F: nat > nat,A3: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( power_power_nat @ ( F @ X4 ) @ N2 )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_6308_prod__power__distrib,axiom,
    ! [F: nat > int,A3: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A3 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( power_power_int @ ( F @ X4 ) @ N2 )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_6309_prod__power__distrib,axiom,
    ! [F: int > int,A3: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( power_power_int @ ( F @ X4 ) @ N2 )
        @ A3 ) ) ).

% prod_power_distrib
thf(fact_6310_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A3: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6311_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A3: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6312_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A3: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A3 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6313_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( G @ zero_zero_nat )
        @ ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6314_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6315_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6316_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6317_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6318_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6319_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U2: set_int] :
          ( collect_set_int
          @ ^ [X4: set_int] : ( ord_less_eq_set_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6320_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_eq_rat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6321_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6322_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6323_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6324_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_rat @ Xs )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_6325_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_6326_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_6327_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_6328_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_6329_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_6330_prod__nonneg,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_6331_prod__mono,axiom,
    ! [A3: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A3 ) @ ( groups7694694392188491536nteger @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6332_prod__mono,axiom,
    ! [A3: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A3 ) @ ( groups3455450783089532116nteger @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6333_prod__mono,axiom,
    ! [A3: set_int,F: int > code_integer,G: int > code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A3 ) @ ( groups3827104343326376752nteger @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6334_prod__mono,axiom,
    ! [A3: set_o,F: $o > rat,G: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A3 ) @ ( groups2869687844427037835_o_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6335_prod__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6336_prod__mono,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6337_prod__mono,axiom,
    ! [A3: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A3 ) @ ( groups3504817904513533571_o_nat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6338_prod__mono,axiom,
    ! [A3: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) @ ( groups1707563613775114915nt_nat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6339_prod__mono,axiom,
    ! [A3: set_o,F: $o > int,G: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
            & ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A3 ) @ ( groups3502327434004483295_o_int @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6340_prod__mono,axiom,
    ! [A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ ( groups708209901874060359at_nat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_6341_prod__pos,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_6342_prod__pos,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_6343_prod__pos,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_6344_prod__ge__1,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6345_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6346_prod__ge__1,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6347_prod__ge__1,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6348_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6349_prod__ge__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6350_prod__ge__1,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups3504817904513533571_o_nat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6351_prod__ge__1,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6352_prod__ge__1,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6353_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_6354_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_6355_not__Iic__le__Icc,axiom,
    ! [H: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_6356_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6357_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6358_power__sum,axiom,
    ! [C: nat,F: nat > nat,A3: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
        @ A3 ) ) ).

% power_sum
thf(fact_6359_power__sum,axiom,
    ! [C: int,F: nat > nat,A3: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A3 ) ) ).

% power_sum
thf(fact_6360_power__sum,axiom,
    ! [C: int,F: int > nat,A3: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A3 ) ) ).

% power_sum
thf(fact_6361_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6362_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6363_prod__le__1,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A3 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6364_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A3 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6365_prod__le__1,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A3 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6366_prod__le__1,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6367_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6368_prod__le__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6369_prod__le__1,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A3 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6370_prod__le__1,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6371_prod__le__1,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) )
            & ( ord_less_eq_int @ ( F @ X3 ) @ one_one_int ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A3 ) @ one_one_int ) ) ).

% prod_le_1
thf(fact_6372_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6373_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P4: int > int > $o,Xs3: list_int] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs3 ) )
           => ( P4 @ ( nth_int @ Xs3 @ I4 ) @ ( nth_int @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_6374_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P4: nat > nat > $o,Xs3: list_nat] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
           => ( P4 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_6375_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_o
    = ( ^ [P4: $o > $o > $o,Xs3: list_o] :
        ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs3 ) )
           => ( P4 @ ( nth_o @ Xs3 @ I4 ) @ ( nth_o @ Xs3 @ J3 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_6376_sorted__wrt__nth__less,axiom,
    ! [P: int > int > $o,Xs: list_int,I2: nat,J: nat] :
      ( ( sorted_wrt_int @ P @ Xs )
     => ( ( ord_less_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_6377_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,I2: nat,J: nat] :
      ( ( sorted_wrt_nat @ P @ Xs )
     => ( ( ord_less_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_6378_sorted__wrt__nth__less,axiom,
    ! [P: $o > $o > $o,Xs: list_o,I2: nat,J: nat] :
      ( ( sorted_wrt_o @ P @ Xs )
     => ( ( ord_less_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_6379_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_6380_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_6381_sorted__wrt01,axiom,
    ! [Xs: list_o,P: $o > $o > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( sorted_wrt_o @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_6382_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6383_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6384_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups3455450783089532116nteger
            @ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_Code_integer @ ( J3 = K ) @ one_one_Code_integer @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3455450783089532116nteger
            @ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6385_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > assn,H: nat > assn] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups6906906614972039071t_assn
            @ ^ [J3: nat] : ( if_assn @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_assn @ ( J3 = K ) @ one_one_assn @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups6906906614972039071t_assn
            @ ^ [J3: nat] : ( if_assn @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6386_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6387_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6388_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6389_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I4 ) @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_6390_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I4 ) @ ( nth_rat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_6391_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I4 ) @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_6392_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_6393_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_6394_sorted01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( sorted_wrt_o @ ord_less_eq_o @ Xs ) ) ).

% sorted01
thf(fact_6395_sorted01,axiom,
    ! [Xs: list_rat] :
      ( ( ord_less_eq_nat @ ( size_size_list_rat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_rat @ ord_less_eq_rat @ Xs ) ) ).

% sorted01
thf(fact_6396_sorted01,axiom,
    ! [Xs: list_num] :
      ( ( ord_less_eq_nat @ ( size_size_list_num @ Xs ) @ one_one_nat )
     => ( sorted_wrt_num @ ord_less_eq_num @ Xs ) ) ).

% sorted01
thf(fact_6397_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_6398_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_6399_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6400_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6401_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6402_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6403_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6404_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_3573771949741848930nteger @ ( G @ ( suc @ N2 ) ) @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6405_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_assn @ ( G @ ( suc @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6406_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6407_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6408_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6409_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_3573771949741848930nteger @ ( G @ M ) @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6410_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_assn @ ( G @ M ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6411_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6412_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6413_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6414_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_3573771949741848930nteger @ ( G @ M )
          @ ( groups3455450783089532116nteger
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6415_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_assn @ ( G @ M )
          @ ( groups6906906614972039071t_assn
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6416_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ M )
          @ ( groups73079841787564623at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6417_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6418_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6419_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6420_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6421_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6422_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6423_sum__telescope,axiom,
    ! [F: nat > rat,I2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_6424_sum__telescope,axiom,
    ! [F: nat > int,I2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_6425_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_6426_fact__prod,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N: nat] :
          ( semiri4939895301339042750nteger
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_6427_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_6428_prod__atLeastAtMost__code,axiom,
    ! [F: nat > code_integer,A: nat,B: nat] :
      ( ( groups3455450783089532116nteger @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A5: nat] : ( times_3573771949741848930nteger @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_Code_integer ) ) ).

% prod_atLeastAtMost_code
thf(fact_6429_prod__atLeastAtMost__code,axiom,
    ! [F: nat > assn,A: nat,B: nat] :
      ( ( groups6906906614972039071t_assn @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1959793692361082170t_assn
        @ ^ [A5: nat] : ( times_times_assn @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_assn ) ) ).

% prod_atLeastAtMost_code
thf(fact_6430_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6431_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6432_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_6433_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_o @ Xs ) )
           => ( ord_less_eq_o @ ( nth_o @ Xs @ I4 ) @ ( nth_o @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_6434_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_rat @ Xs ) )
           => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I4 ) @ ( nth_rat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_6435_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_num @ Xs ) )
           => ( ord_less_eq_num @ ( nth_num @ Xs @ I4 ) @ ( nth_num @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_6436_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_6437_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_6438_sorted__nth__mono,axiom,
    ! [Xs: list_o,I2: nat,J: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
         => ( ord_less_eq_o @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_6439_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I2: nat,J: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_rat @ Xs ) )
         => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I2 ) @ ( nth_rat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_6440_sorted__nth__mono,axiom,
    ! [Xs: list_num,I2: nat,J: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs ) )
         => ( ord_less_eq_num @ ( nth_num @ Xs @ I2 ) @ ( nth_num @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_6441_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I2: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_6442_sorted__nth__mono,axiom,
    ! [Xs: list_int,I2: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_6443_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I4 ) @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_6444_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I4 ) @ ( nth_rat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_6445_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I4 ) @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_6446_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_6447_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I4: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I4 @ J3 )
           => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I4 ) @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_6448_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6449_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > assn,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6450_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6451_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6452_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P3: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P3 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P3 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6453_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_6454_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_6455_sum__gp__basic,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6456_sum__gp__basic,axiom,
    ! [X: rat,N2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6457_sum__gp__basic,axiom,
    ! [X: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6458_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6459_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6460_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6461_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6462_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6463_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6464_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6465_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6466_pochhammer__prod__rev,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6467_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6468_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6469_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_6470_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_6471_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6472_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N2 @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6473_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6474_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6475_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups7501900531339628137nteger
            @ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_Code_integer @ ( J3 = K ) @ zero_z3403309356797280102nteger @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups7501900531339628137nteger
            @ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6476_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > multis2468970476368604999at_nat,H: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups6857163185585827899at_nat
            @ ^ [J3: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_mul8430962117462786573at_nat @ ( J3 = K ) @ zero_z1048942125864253310at_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups6857163185585827899at_nat
            @ ^ [J3: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6477_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6478_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6479_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K @ P3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6480_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_6481_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_6482_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_6483_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_6484_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_6485_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_6486_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6487_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > rat,D2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D2 @ I4 ) )
            @ A3 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D2 @ I4 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_6488_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_6489_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_6490_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N2 @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_6491_gbinomial__prod__rev,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( divide_divide_rat
          @ ( groups73079841787564623at_rat
            @ ^ [I4: nat] : ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ I4 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_6492_gbinomial__prod__rev,axiom,
    ( gbinom7368847122466276068atural
    = ( ^ [A5: code_natural,K3: nat] :
          ( divide5121882707175180666atural
          @ ( groups2279045934846249631atural
            @ ^ [I4: nat] : ( minus_7197305767214868737atural @ A5 @ ( semiri3763490453095760265atural @ I4 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri2447717529341329178atural @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_6493_gbinomial__prod__rev,axiom,
    ( gbinom8545251970709558553nteger
    = ( ^ [A5: code_integer,K3: nat] :
          ( divide6298287555418463151nteger
          @ ( groups3455450783089532116nteger
            @ ^ [I4: nat] : ( minus_8373710615458151222nteger @ A5 @ ( semiri4939895301339042750nteger @ I4 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri3624122377584611663nteger @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_6494_gbinomial__prod__rev,axiom,
    ( gbinomial_nat
    = ( ^ [A5: nat,K3: nat] :
          ( divide_divide_nat
          @ ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ A5 @ ( semiri1316708129612266289at_nat @ I4 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_6495_gbinomial__prod__rev,axiom,
    ( gbinomial_int
    = ( ^ [A5: int,K3: nat] :
          ( divide_divide_int
          @ ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( minus_minus_int @ A5 @ ( semiri1314217659103216013at_int @ I4 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri1406184849735516958ct_int @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_6496_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_6497_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(71)
thf(fact_6498_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_6499_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_6500_atLeastLessThan__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( ( member_o @ I2 @ ( set_or7139685690850216873Than_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I2 )
        & ( ord_less_o @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6501_atLeastLessThan__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or3540276404033026485et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I2 )
        & ( ord_less_set_nat @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6502_atLeastLessThan__iff,axiom,
    ! [I2: assn,L: assn,U: assn] :
      ( ( member_assn @ I2 @ ( set_or5523502641901747543n_assn @ L @ U ) )
      = ( ( ord_less_eq_assn @ L @ I2 )
        & ( ord_less_assn @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6503_atLeastLessThan__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I2 @ ( set_or8585797421378605585et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I2 )
        & ( ord_less_set_int @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6504_atLeastLessThan__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( ( member_rat @ I2 @ ( set_or4029947393144176647an_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I2 )
        & ( ord_less_rat @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6505_atLeastLessThan__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( ( member_num @ I2 @ ( set_or1222409239386451017an_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I2 )
        & ( ord_less_num @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6506_atLeastLessThan__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_nat @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6507_atLeastLessThan__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( ( member_int @ I2 @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I2 )
        & ( ord_less_int @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6508_atLeastLessThan__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I2 @ ( set_or8404916559141939852nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I2 )
        & ( ord_le6747313008572928689nteger @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_6509_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_eq_o @ B @ A )
     => ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_6510_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( set_or8585797421378605585et_int @ A @ B )
        = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_6511_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
thf(fact_6512_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
thf(fact_6513_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_6514_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_6515_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_6516_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_6517_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A @ B ) )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6518_atLeastLessThan__empty__iff2,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_set_assn
        = ( set_or5523502641901747543n_assn @ A @ B ) )
      = ( ~ ( ord_less_assn @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6519_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or4029947393144176647an_rat @ A @ B ) )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6520_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or1222409239386451017an_num @ A @ B ) )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6521_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6522_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A @ B ) )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6523_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A @ B ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6524_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6525_atLeastLessThan__empty__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( ( set_or5523502641901747543n_assn @ A @ B )
        = bot_bot_set_assn )
      = ( ~ ( ord_less_assn @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6526_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6527_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6528_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6529_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6530_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6531_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_6532_ivl__subset,axiom,
    ! [I2: rat,J: rat,M: rat,N2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I2 @ J ) @ ( set_or4029947393144176647an_rat @ M @ N2 ) )
      = ( ( ord_less_eq_rat @ J @ I2 )
        | ( ( ord_less_eq_rat @ M @ I2 )
          & ( ord_less_eq_rat @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_6533_ivl__subset,axiom,
    ! [I2: num,J: num,M: num,N2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I2 @ J ) @ ( set_or1222409239386451017an_num @ M @ N2 ) )
      = ( ( ord_less_eq_num @ J @ I2 )
        | ( ( ord_less_eq_num @ M @ I2 )
          & ( ord_less_eq_num @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_6534_ivl__subset,axiom,
    ! [I2: nat,J: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ J @ I2 )
        | ( ( ord_less_eq_nat @ M @ I2 )
          & ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_6535_ivl__subset,axiom,
    ! [I2: int,J: int,M: int,N2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I2 @ J ) @ ( set_or4662586982721622107an_int @ M @ N2 ) )
      = ( ( ord_less_eq_int @ J @ I2 )
        | ( ( ord_less_eq_int @ M @ I2 )
          & ( ord_less_eq_int @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_6536_ivl__subset,axiom,
    ! [I2: code_integer,J: code_integer,M: code_integer,N2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I2 @ J ) @ ( set_or8404916559141939852nteger @ M @ N2 ) )
      = ( ( ord_le3102999989581377725nteger @ J @ I2 )
        | ( ( ord_le3102999989581377725nteger @ M @ I2 )
          & ( ord_le3102999989581377725nteger @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_6537_ivl__diff,axiom,
    ! [I2: rat,N2: rat,M: rat] :
      ( ( ord_less_eq_rat @ I2 @ N2 )
     => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I2 @ M ) @ ( set_or4029947393144176647an_rat @ I2 @ N2 ) )
        = ( set_or4029947393144176647an_rat @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_6538_ivl__diff,axiom,
    ! [I2: num,N2: num,M: num] :
      ( ( ord_less_eq_num @ I2 @ N2 )
     => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I2 @ M ) @ ( set_or1222409239386451017an_num @ I2 @ N2 ) )
        = ( set_or1222409239386451017an_num @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_6539_ivl__diff,axiom,
    ! [I2: nat,N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ N2 )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M ) @ ( set_or4665077453230672383an_nat @ I2 @ N2 ) )
        = ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_6540_ivl__diff,axiom,
    ! [I2: int,N2: int,M: int] :
      ( ( ord_less_eq_int @ I2 @ N2 )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I2 @ M ) @ ( set_or4662586982721622107an_int @ I2 @ N2 ) )
        = ( set_or4662586982721622107an_int @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_6541_ivl__diff,axiom,
    ! [I2: code_integer,N2: code_integer,M: code_integer] :
      ( ( ord_le3102999989581377725nteger @ I2 @ N2 )
     => ( ( minus_2355218937544613996nteger @ ( set_or8404916559141939852nteger @ I2 @ M ) @ ( set_or8404916559141939852nteger @ I2 @ N2 ) )
        = ( set_or8404916559141939852nteger @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_6542_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_6543_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_6544_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_6545_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups4406642042086082107nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups4406642042086082107nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta'
thf(fact_6546_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups7501900531339628137nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups7501900531339628137nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta'
thf(fact_6547_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7873554091576472773nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7873554091576472773nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta'
thf(fact_6548_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups879477027807139574nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups879477027807139574nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta'
thf(fact_6549_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6550_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6551_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6552_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6553_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_6554_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_6555_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups4406642042086082107nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups4406642042086082107nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta
thf(fact_6556_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups7501900531339628137nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups7501900531339628137nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta
thf(fact_6557_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7873554091576472773nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7873554091576472773nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta
thf(fact_6558_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups879477027807139574nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups879477027807139574nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ zero_z3403309356797280102nteger )
              @ S )
            = zero_z3403309356797280102nteger ) ) ) ) ).

% sum.delta
thf(fact_6559_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6560_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6561_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6562_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6563_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_6564_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_6565_image__add__atLeastLessThan_H,axiom,
    ! [K: rat,I2: rat,J: rat] :
      ( ( image_rat_rat
        @ ^ [N: rat] : ( plus_plus_rat @ N @ K )
        @ ( set_or4029947393144176647an_rat @ I2 @ J ) )
      = ( set_or4029947393144176647an_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_6566_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I2: nat,J: nat] :
      ( ( image_nat_nat
        @ ^ [N: nat] : ( plus_plus_nat @ N @ K )
        @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_6567_image__add__atLeastLessThan_H,axiom,
    ! [K: int,I2: int,J: int] :
      ( ( image_int_int
        @ ^ [N: int] : ( plus_plus_int @ N @ K )
        @ ( set_or4662586982721622107an_int @ I2 @ J ) )
      = ( set_or4662586982721622107an_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_6568_image__add__atLeastLessThan_H,axiom,
    ! [K: code_integer,I2: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [N: code_integer] : ( plus_p5714425477246183910nteger @ N @ K )
        @ ( set_or8404916559141939852nteger @ I2 @ J ) )
      = ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_6569_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups7694694392188491536nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups7694694392188491536nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_6570_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3455450783089532116nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3455450783089532116nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_6571_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups3827104343326376752nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups3827104343326376752nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_6572_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups3674199335183972705nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups3674199335183972705nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( A = K3 ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_6573_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > assn] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_6574_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > assn] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_6575_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > assn] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_6576_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( A = K3 ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_6577_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_6578_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_6579_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups7694694392188491536nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups7694694392188491536nteger
              @ ^ [K3: $o] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_6580_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3455450783089532116nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3455450783089532116nteger
              @ ^ [K3: nat] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_6581_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups3827104343326376752nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups3827104343326376752nteger
              @ ^ [K3: int] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_6582_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups3674199335183972705nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups3674199335183972705nteger
              @ ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = A ) @ ( B @ K3 ) @ one_one_Code_integer )
              @ S )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_6583_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > assn] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_6584_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > assn] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_6585_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: int > assn] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_6586_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ one_one_assn )
              @ S )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_6587_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_6588_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_6589_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_6590_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6591_prod__pos__nat__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups3190895334310489300er_nat @ F @ A3 ) )
        = ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6592_prod__pos__nat__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A3 ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6593_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6594_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6595_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6596_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6597_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6598_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6599_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6600_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_6601_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_6602_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_6603_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_6604_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_6605_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_6606_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_6607_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_6608_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_6609_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_6610_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_6611_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_6612_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_6613_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6614_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6615_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6616_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_6617_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_6618_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_6619_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_6620_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_6621_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_6622_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_6623_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_6624_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_6625_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6626_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6627_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6628_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6629_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6630_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6631_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_6632_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_6633_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_6634_int__prod,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% int_prod
thf(fact_6635_int__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A3 ) ) ).

% int_prod
thf(fact_6636_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S4: set_int] :
      ( ( finite_finite_int @ S )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S6: set_int] :
              ( ( minus_minus_set_int @ S6 @ S )
              = ( minus_minus_set_int @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6637_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S6: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S6 @ S )
              = ( minus_2355218937544613996nteger @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6638_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [S6: set_Pr1261947904930325089at_nat] :
              ( ( minus_1356011639430497352at_nat @ S6 @ S )
              = ( minus_1356011639430497352at_nat @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6639_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S4: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S6: set_nat] :
              ( ( minus_minus_set_nat @ S6 @ S )
              = ( minus_minus_set_nat @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_6640_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_6641_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_6642_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_6643_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C @ D2 )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_6644_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_6645_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_6646_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_6647_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_6648_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_6649_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_6650_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_6651_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_6652_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_6653_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_6654_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_6655_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_6656_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_6657_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_6658_choose__row__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% choose_row_sum
thf(fact_6659_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M3: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N8 )
         => ( ord_less_nat @ X4 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_6660_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N7 )
         => ( ord_less_nat @ X3 @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_6661_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M3: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N8 )
         => ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_6662_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% choose_square_sum
thf(fact_6663_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_6664_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_6665_mult__2,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2
thf(fact_6666_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_6667_mult__2,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2
thf(fact_6668_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6669_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6670_mult__2__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6671_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6672_mult__2__right,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ Z @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6673_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6674_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6675_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6676_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6677_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_6678_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_6679_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_6680_power4__eq__xxxx,axiom,
    ! [X: assn] :
      ( ( power_power_assn @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_6681_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_6682_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_6683_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_6684_power2__eq__square,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% power2_eq_square
thf(fact_6685_power2__eq__square,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_assn @ A @ A ) ) ).

% power2_eq_square
thf(fact_6686_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_6687_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_6688_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_6689_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_6690_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_6691_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: $o > nat > nat,R: $o > nat > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups8507830703676809646_o_nat
            @ ^ [X4: $o] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups8507830703676809646_o_nat
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6692_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6693_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: code_integer > nat > nat,R: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X4: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6694_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_int,G: $o > int > int,R: $o > int > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups8505340233167759370_o_int
            @ ^ [X4: $o] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups8505340233167759370_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6695_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: nat > int > int,R: nat > int > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X4: nat] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups3539618377306564664at_int
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6696_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_int,G: code_integer > int > int,R: code_integer > int > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups7234854612051535045er_int
            @ ^ [X4: code_integer] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups7234854612051535045er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6697_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_o,G: nat > $o > nat,R: nat > $o > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups8507830703676809646_o_nat @ ( G @ X4 )
                @ ( collect_o
                  @ ^ [Y4: $o] :
                      ( ( member_o @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups8507830703676809646_o_nat
            @ ^ [Y4: $o] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6698_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: nat > int > nat,R: nat > int > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups4541462559716669496nt_nat @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups4541462559716669496nt_nat
            @ ^ [Y4: int] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6699_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_Code_integer,G: nat > code_integer > nat,R: nat > code_integer > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups7237345082560585321er_nat @ ( G @ X4 )
                @ ( collect_Code_integer
                  @ ^ [Y4: code_integer] :
                      ( ( member_Code_integer @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups7237345082560585321er_nat
            @ ^ [Y4: code_integer] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6700_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > nat > nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6701_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: $o > nat > nat,R: $o > nat > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3504817904513533571_o_nat
            @ ^ [X4: $o] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups3504817904513533571_o_nat
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6702_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B3: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups1707563613775114915nt_nat
            @ ^ [X4: int] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups1707563613775114915nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6703_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: code_integer > nat > nat,R: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3190895334310489300er_nat
            @ ^ [X4: code_integer] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups3190895334310489300er_nat
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6704_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_nat,G: $o > nat > int,R: $o > nat > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X4: $o] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups705719431365010083at_int
            @ ^ [Y4: nat] :
                ( groups3502327434004483295_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6705_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_nat,G: code_integer > nat > int,R: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X4: code_integer] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups705719431365010083at_int
            @ ^ [Y4: nat] :
                ( groups3188404863801439024er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6706_prod_Oswap__restrict,axiom,
    ! [A3: set_o,B3: set_int,G: $o > int > int,R: $o > int > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X4: $o] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y4: int] :
                ( groups3502327434004483295_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6707_prod_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B3: set_int,G: code_integer > int > int,R: code_integer > int > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X4: code_integer] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y4: int] :
                ( groups3188404863801439024er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6708_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_o,G: nat > $o > nat,R: nat > $o > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups3504817904513533571_o_nat @ ( G @ X4 )
                @ ( collect_o
                  @ ^ [Y4: $o] :
                      ( ( member_o @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3504817904513533571_o_nat
            @ ^ [Y4: $o] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6709_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_int,G: nat > int > nat,R: nat > int > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups1707563613775114915nt_nat @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups1707563613775114915nt_nat
            @ ^ [Y4: int] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6710_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_Code_integer,G: nat > code_integer > nat,R: nat > code_integer > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups3190895334310489300er_nat @ ( G @ X4 )
                @ ( collect_Code_integer
                  @ ^ [Y4: code_integer] :
                      ( ( member_Code_integer @ Y4 @ B3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ A3 )
          = ( groups3190895334310489300er_nat
            @ ^ [Y4: code_integer] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A3 )
                      & ( R @ X4 @ Y4 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_6711_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_6712_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_6713_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_6714_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_6715_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_6716_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_6717_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6718_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_6719_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_6720_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ B @ A )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_6721_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_6722_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_6723_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_6724_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_6725_zero__le__power2,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_6726_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_6727_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_6728_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6729_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6730_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6731_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_6732_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6733_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6734_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6735_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_6736_power2__less__0,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_6737_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_6738_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_6739_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_6740_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_eq_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_6741_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_eq_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ M ) @ ( set_or1222409239386451017an_num @ M @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_6742_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_eq_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_6743_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_eq_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_6744_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M )
     => ( ( ord_le3102999989581377725nteger @ M @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_6745_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
            & ( P @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_6746_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
           => ( P @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_6747_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_6748_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_6749_ivl__disj__int__two_I3_J,axiom,
    ! [L: $o,M: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ M ) @ ( set_or7139685690850216873Than_o @ M @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_two(3)
thf(fact_6750_ivl__disj__int__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_two(3)
thf(fact_6751_ivl__disj__int__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_two(3)
thf(fact_6752_ivl__disj__int__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( inf_in1364745209274528805nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
      = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_two(3)
thf(fact_6753_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6754_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6755_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6756_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6757_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ S )
            & ~ ? [Xa2: $o] :
                  ( ( member_o @ Xa2 @ S )
                  & ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6758_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( ( finite_finite_assn @ S )
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( ( member_assn @ X3 @ S )
            & ~ ? [Xa2: assn] :
                  ( ( member_assn @ Xa2 @ S )
                  & ( ord_less_assn @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6759_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( ( finite_finite_rat @ S )
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S )
            & ~ ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ S )
                  & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6760_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( ( finite_finite_num @ S )
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S )
            & ~ ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ S )
                  & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6761_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6762_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S )
                  & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6763_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X6 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X6 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X6 ) ) ) ).

% infinite_growing
thf(fact_6764_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X6 )
           => ? [Xa2: $o] :
                ( ( member_o @ Xa2 @ X6 )
                & ( ord_less_o @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_o @ X6 ) ) ) ).

% infinite_growing
thf(fact_6765_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X6 )
           => ? [Xa2: rat] :
                ( ( member_rat @ Xa2 @ X6 )
                & ( ord_less_rat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_rat @ X6 ) ) ) ).

% infinite_growing
thf(fact_6766_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X6 )
           => ? [Xa2: num] :
                ( ( member_num @ Xa2 @ X6 )
                & ( ord_less_num @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_num @ X6 ) ) ) ).

% infinite_growing
thf(fact_6767_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X6 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_6768_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X6 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X6 )
                & ( ord_less_int @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_6769_binomial__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R3 ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_6770_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_6771_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_6772_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_6773_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N2 @ I4 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_6774_sum__mono__inv,axiom,
    ! [F: $o > rat,I5: set_o,G: $o > rat,I2: $o] :
      ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
        = ( groups7872700643590313910_o_rat @ G @ I5 ) )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o @ I2 @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6775_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I2: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I2 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6776_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I2: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I2 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6777_sum__mono__inv,axiom,
    ! [F: code_integer > rat,I5: set_Code_integer,G: code_integer > rat,I2: code_integer] :
      ( ( ( groups6602215022474089585er_rat @ F @ I5 )
        = ( groups6602215022474089585er_rat @ G @ I5 ) )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Code_integer @ I2 @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6778_sum__mono__inv,axiom,
    ! [F: $o > nat,I5: set_o,G: $o > nat,I2: $o] :
      ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
        = ( groups8507830703676809646_o_nat @ G @ I5 ) )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o @ I2 @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6779_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I2: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I2 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6780_sum__mono__inv,axiom,
    ! [F: code_integer > nat,I5: set_Code_integer,G: code_integer > nat,I2: code_integer] :
      ( ( ( groups7237345082560585321er_nat @ F @ I5 )
        = ( groups7237345082560585321er_nat @ G @ I5 ) )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Code_integer @ I2 @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6781_sum__mono__inv,axiom,
    ! [F: $o > int,I5: set_o,G: $o > int,I2: $o] :
      ( ( ( groups8505340233167759370_o_int @ F @ I5 )
        = ( groups8505340233167759370_o_int @ G @ I5 ) )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o @ I2 @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6782_sum__mono__inv,axiom,
    ! [F: nat > int,I5: set_nat,G: nat > int,I2: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I5 )
        = ( groups3539618377306564664at_int @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I2 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6783_sum__mono__inv,axiom,
    ! [F: code_integer > int,I5: set_Code_integer,G: code_integer > int,I2: code_integer] :
      ( ( ( groups7234854612051535045er_int @ F @ I5 )
        = ( groups7234854612051535045er_int @ G @ I5 ) )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Code_integer @ I2 @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6784_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_6785_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6786_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6787_mult__numeral__1__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6788_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6789_mult__numeral__1__right,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_6790_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6791_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6792_mult__numeral__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6793_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6794_mult__numeral__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_6795_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_6796_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_6797_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_6798_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_6799_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_6800_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6801_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6802_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6803_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6804_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6805_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6806_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6807_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > rat,Y: code_integer > rat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6808_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > nat,Y: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6809_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_nat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6810_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_6811_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_6812_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6813_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6814_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6815_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6816_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6817_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6818_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_Code_integer ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6819_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > assn,Y: $o > assn] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_assn @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6820_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > assn,Y: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_assn @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6821_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > assn,Y: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_assn @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6822_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > assn,Y: code_integer > assn] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_assn ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_assn ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( times_times_assn @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6823_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6824_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X @ I4 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_rat @ ( X @ I4 ) @ ( Y @ I4 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6825_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6826_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6827_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6828_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_6829_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6830_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6831_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_6832_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6833_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6834_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_6835_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6836_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6837_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_6838_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_6839_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_6840_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_6841_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6842_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6843_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6844_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6845_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6846_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > code_integer,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups4406642042086082107nteger @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups4406642042086082107nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6847_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > code_integer,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups7501900531339628137nteger @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6848_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > code_integer,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7873554091576472773nteger @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6849_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups879477027807139574nteger @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6850_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > rat,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7872700643590313910_o_rat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6851_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6852_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6853_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6854_sum_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > nat,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups8507830703676809646_o_nat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6855_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > nat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6856_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6857_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6858_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6859_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6860_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( power_7079662738309270450atural @ ( plus_p4538020629002901425atural @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6861_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_6862_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_6863_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_6864_filter__preserves__multiset,axiom,
    ! [M6: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6865_filter__preserves__multiset,axiom,
    ! [M6: list_nat > nat,P: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6866_filter__preserves__multiset,axiom,
    ! [M6: set_nat > nat,P: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6867_filter__preserves__multiset,axiom,
    ! [M6: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6868_filter__preserves__multiset,axiom,
    ! [M6: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6869_filter__preserves__multiset,axiom,
    ! [M6: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6870_filter__preserves__multiset,axiom,
    ! [M6: product_prod_nat_nat > nat,P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_6871_zero__le__even__power_H,axiom,
    ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_6872_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_6873_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_6874_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6875_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6876_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6877_sum_Oimage__gen,axiom,
    ! [S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( groups8507830703676809646_o_nat @ H @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups8507830703676809646_o_nat @ H
              @ ( collect_o
                @ ^ [X4: $o] :
                    ( ( member_o @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_o_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6878_sum_Oimage__gen,axiom,
    ! [S: set_int,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( groups4541462559716669496nt_nat @ H @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups4541462559716669496nt_nat @ H
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6879_sum_Oimage__gen,axiom,
    ! [S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( groups7237345082560585321er_nat @ H @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups7237345082560585321er_nat @ H
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_951025933927791156er_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6880_sum_Oimage__gen,axiom,
    ! [S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( groups8505340233167759370_o_int @ H @ S )
        = ( groups4538972089207619220nt_int
          @ ^ [Y4: int] :
              ( groups8505340233167759370_o_int @ H
              @ ( collect_o
                @ ^ [X4: $o] :
                    ( ( member_o @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_o_int @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6881_sum_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( groups3539618377306564664at_int @ H @ S )
        = ( groups4538972089207619220nt_int
          @ ^ [Y4: int] :
              ( groups3539618377306564664at_int @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_int @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6882_sum_Oimage__gen,axiom,
    ! [S: set_Code_integer,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( groups7234854612051535045er_int @ H @ S )
        = ( groups4538972089207619220nt_int
          @ ^ [Y4: int] :
              ( groups7234854612051535045er_int @ H
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_948535463418740880er_int @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6883_sum_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > nat,G: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( groups3542108847815614940at_nat @ H @ S )
        = ( groups4541462559716669496nt_nat
          @ ^ [Y4: int] :
              ( groups3542108847815614940at_nat @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_int @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6884_sum_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( groups3542108847815614940at_nat @ H @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y4: nat] :
              ( groups3542108847815614940at_nat @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6885_sum_Oimage__gen,axiom,
    ! [S: set_int,H: int > int,G: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( groups4538972089207619220nt_int @ H @ S )
        = ( groups3539618377306564664at_int
          @ ^ [Y4: nat] :
              ( groups4538972089207619220nt_int @ H
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6886_sum_Oimage__gen,axiom,
    ! [S: set_int,H: int > int,G: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( groups4538972089207619220nt_int @ H @ S )
        = ( groups4538972089207619220nt_int
          @ ^ [Y4: int] :
              ( groups4538972089207619220nt_int @ H
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_int @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_6887_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6888_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6889_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6890_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6891_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6892_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6893_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > code_integer,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7694694392188491536nteger @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups7694694392188491536nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6894_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > code_integer,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups3455450783089532116nteger
          @ ^ [X4: nat] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6895_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: int > code_integer,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups3827104343326376752nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6896_prod_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups3674199335183972705nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6897_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > assn,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups5301882518646026715o_assn @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups5301882518646026715o_assn
          @ ^ [X4: $o] : ( if_assn @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6898_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > assn,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups6906906614972039071t_assn @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups6906906614972039071t_assn
          @ ^ [X4: nat] : ( if_assn @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6899_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: int > assn,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6900_prod_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: code_integer > assn,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6901_prod_Ointer__filter,axiom,
    ! [A3: set_o,G: $o > rat,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups2869687844427037835_o_rat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups2869687844427037835_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6902_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
                & ( P @ X4 ) ) ) )
        = ( groups73079841787564623at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_6903_prod_Oimage__gen,axiom,
    ! [S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( groups3504817904513533571_o_nat @ H @ S )
        = ( groups708209901874060359at_nat
          @ ^ [Y4: nat] :
              ( groups3504817904513533571_o_nat @ H
              @ ( collect_o
                @ ^ [X4: $o] :
                    ( ( member_o @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_o_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6904_prod_Oimage__gen,axiom,
    ! [S: set_int,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( groups1707563613775114915nt_nat @ H @ S )
        = ( groups708209901874060359at_nat
          @ ^ [Y4: nat] :
              ( groups1707563613775114915nt_nat @ H
              @ ( collect_int
                @ ^ [X4: int] :
                    ( ( member_int @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_int_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6905_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( groups3190895334310489300er_nat @ H @ S )
        = ( groups708209901874060359at_nat
          @ ^ [Y4: nat] :
              ( groups3190895334310489300er_nat @ H
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_951025933927791156er_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6906_prod_Oimage__gen,axiom,
    ! [S: set_o,H: $o > int,G: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( groups3502327434004483295_o_int @ H @ S )
        = ( groups705719431365010083at_int
          @ ^ [Y4: nat] :
              ( groups3502327434004483295_o_int @ H
              @ ( collect_o
                @ ^ [X4: $o] :
                    ( ( member_o @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_o_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6907_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,H: code_integer > int,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( groups3188404863801439024er_int @ H @ S )
        = ( groups705719431365010083at_int
          @ ^ [Y4: nat] :
              ( groups3188404863801439024er_int @ H
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_951025933927791156er_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6908_prod_Oimage__gen,axiom,
    ! [S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( groups3502327434004483295_o_int @ H @ S )
        = ( groups1705073143266064639nt_int
          @ ^ [Y4: int] :
              ( groups3502327434004483295_o_int @ H
              @ ( collect_o
                @ ^ [X4: $o] :
                    ( ( member_o @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_o_int @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6909_prod_Oimage__gen,axiom,
    ! [S: set_Code_integer,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( groups3188404863801439024er_int @ H @ S )
        = ( groups1705073143266064639nt_int
          @ ^ [Y4: int] :
              ( groups3188404863801439024er_int @ H
              @ ( collect_Code_integer
                @ ^ [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_948535463418740880er_int @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6910_prod_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > nat,G: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( groups708209901874060359at_nat @ H @ S )
        = ( groups1707563613775114915nt_nat
          @ ^ [Y4: int] :
              ( groups708209901874060359at_nat @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_int @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6911_prod_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( groups708209901874060359at_nat @ H @ S )
        = ( groups708209901874060359at_nat
          @ ^ [Y4: nat] :
              ( groups708209901874060359at_nat @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6912_prod_Oimage__gen,axiom,
    ! [S: set_nat,H: nat > int,G: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( groups705719431365010083at_int @ H @ S )
        = ( groups705719431365010083at_int
          @ ^ [Y4: nat] :
              ( groups705719431365010083at_int @ H
              @ ( collect_nat
                @ ^ [X4: nat] :
                    ( ( member_nat @ X4 @ S )
                    & ( ( G @ X4 )
                      = Y4 ) ) ) )
          @ ( image_nat_nat @ G @ S ) ) ) ) ).

% prod.image_gen
thf(fact_6913_minus__power__mult__self,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_6914_minus__power__mult__self,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_6915_minus__power__mult__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_6916_power__odd__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_6917_power__odd__eq,axiom,
    ! [A: assn,N2: nat] :
      ( ( power_power_assn @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_assn @ A @ ( power_power_assn @ ( power_power_assn @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_6918_power__odd__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_6919_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_6920_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_6921_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_6922_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_6923_finite__int__segment,axiom,
    ! [A: rat,B: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [X4: rat] :
            ( ( member_rat @ X4 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ A @ X4 )
            & ( ord_less_eq_rat @ X4 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_6924_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_6925_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > nat,H: nat > nat] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_6926_sum_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D2: int,G: int > int,H: int > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C @ X3 )
             => ( ( ord_less_int @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups4538972089207619220nt_int @ H @ ( set_or4662586982721622107an_int @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_6927_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > nat,H: nat > nat] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups708209901874060359at_nat @ H @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_6928_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > int,H: nat > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups705719431365010083at_int @ H @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_6929_prod_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D2: int,G: int > int,H: int > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C @ X3 )
             => ( ( ord_less_int @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups1705073143266064639nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups1705073143266064639nt_int @ H @ ( set_or4662586982721622107an_int @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_6930_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_6931_ivl__disj__un__two_I7_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_eq_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_6932_ivl__disj__un__two_I7_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_eq_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ M ) @ ( set_or7049704709247886629st_num @ M @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_6933_ivl__disj__un__two_I7_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_eq_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_6934_ivl__disj__un__two_I7_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_eq_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_6935_ivl__disj__un__two_I7_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M )
     => ( ( ord_le3102999989581377725nteger @ M @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
          = ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_6936_binomial__altdef__of__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
        = ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K @ I4 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% binomial_altdef_of_nat
thf(fact_6937_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6938_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6939_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6940_ivl__disj__int__two_I7_J,axiom,
    ! [L: $o,M: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ M ) @ ( set_or8904488021354931149Most_o @ M @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_two(7)
thf(fact_6941_ivl__disj__int__two_I7_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_two(7)
thf(fact_6942_ivl__disj__int__two_I7_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_two(7)
thf(fact_6943_ivl__disj__int__two_I7_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( inf_in1364745209274528805nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
      = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_two(7)
thf(fact_6944_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_6945_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_6946_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_6947_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_6948_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P3: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) )
          = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_6949_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P3: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) )
          = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_6950_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_6951_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_6952_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_6953_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_6954_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6955_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6956_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6957_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6958_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P3 )
       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P3 ) ) )
          = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P3 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6959_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_6960_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_6961_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_6962_odd__power__less__zero,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_6963_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_6964_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_6965_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups4406642042086082107nteger @ F @ A3 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6966_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ A3 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6967_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ A3 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6968_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups879477027807139574nteger @ F @ A3 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6969_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6970_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6971_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6972_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups6602215022474089585er_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6973_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ A3 )
            = zero_zero_nat )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6974_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
            = zero_zero_nat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A3 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6975_sum__le__included,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > code_integer,I2: nat > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6976_sum__le__included,axiom,
    ! [S2: set_nat,T5: set_int,G: int > code_integer,I2: int > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6977_sum__le__included,axiom,
    ! [S2: set_nat,T5: set_Code_integer,G: code_integer > code_integer,I2: code_integer > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6978_sum__le__included,axiom,
    ! [S2: set_int,T5: set_nat,G: nat > code_integer,I2: nat > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6979_sum__le__included,axiom,
    ! [S2: set_int,T5: set_int,G: int > code_integer,I2: int > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6980_sum__le__included,axiom,
    ! [S2: set_int,T5: set_Code_integer,G: code_integer > code_integer,I2: code_integer > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6981_sum__le__included,axiom,
    ! [S2: set_Code_integer,T5: set_nat,G: nat > code_integer,I2: nat > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6982_sum__le__included,axiom,
    ! [S2: set_Code_integer,T5: set_int,G: int > code_integer,I2: int > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6983_sum__le__included,axiom,
    ! [S2: set_Code_integer,T5: set_Code_integer,G: code_integer > code_integer,I2: code_integer > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6984_sum__le__included,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T5 )
                    & ( ( I2 @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T5 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6985_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: nat] :
              ( ( member_nat @ X7 @ A3 )
              & ( ord_less_rat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6986_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: int] :
              ( ( member_int @ X7 @ A3 )
              & ( ord_less_rat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6987_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: code_integer] :
              ( ( member_Code_integer @ X7 @ A3 )
              & ( ord_less_rat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6988_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: int] :
              ( ( member_int @ X7 @ A3 )
              & ( ord_less_nat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6989_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: code_integer] :
              ( ( member_Code_integer @ X7 @ A3 )
              & ( ord_less_nat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6990_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: nat] :
              ( ( member_nat @ X7 @ A3 )
              & ( ord_less_int @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6991_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: code_integer] :
              ( ( member_Code_integer @ X7 @ A3 )
              & ( ord_less_int @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6992_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: nat] :
              ( ( member_nat @ X7 @ A3 )
              & ( ord_less_nat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6993_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A3 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: int] :
              ( ( member_int @ X7 @ A3 )
              & ( ord_less_int @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6994_sum__strict__mono__ex1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X7: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X7 @ A3 )
              & ( ord_less_rat @ ( F @ X7 ) @ ( G @ X7 ) ) )
         => ( ord_less_rat @ ( groups342789780944988191at_rat @ F @ A3 ) @ ( groups342789780944988191at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6995_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6996_sum__strict__mono,axiom,
    ! [A3: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6997_sum__strict__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6998_sum__strict__mono,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6999_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7000_sum__strict__mono,axiom,
    ! [A3: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7001_sum__strict__mono,axiom,
    ! [A3: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7002_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7003_sum__strict__mono,axiom,
    ! [A3: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A3 ) @ ( groups8505340233167759370_o_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7004_sum__strict__mono,axiom,
    ! [A3: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7005_prod_Orelated,axiom,
    ! [R: code_integer > code_integer > $o,S: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( R @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3455450783089532116nteger @ H @ S ) @ ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7006_prod_Orelated,axiom,
    ! [R: code_integer > code_integer > $o,S: set_int,H: int > code_integer,G: int > code_integer] :
      ( ( R @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3827104343326376752nteger @ H @ S ) @ ( groups3827104343326376752nteger @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7007_prod_Orelated,axiom,
    ! [R: code_integer > code_integer > $o,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( R @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3674199335183972705nteger @ H @ S ) @ ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7008_prod_Orelated,axiom,
    ! [R: assn > assn > $o,S: set_nat,H: nat > assn,G: nat > assn] :
      ( ( R @ one_one_assn @ one_one_assn )
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups6906906614972039071t_assn @ H @ S ) @ ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7009_prod_Orelated,axiom,
    ! [R: assn > assn > $o,S: set_int,H: int > assn,G: int > assn] :
      ( ( R @ one_one_assn @ one_one_assn )
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups7882442080178216443t_assn @ H @ S ) @ ( groups7882442080178216443t_assn @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7010_prod_Orelated,axiom,
    ! [R: assn > assn > $o,S: set_Code_integer,H: code_integer > assn,G: code_integer > assn] :
      ( ( R @ one_one_assn @ one_one_assn )
     => ( ! [X12: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1304777262505850412r_assn @ H @ S ) @ ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7011_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_nat,H: nat > rat,G: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7012_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_int,H: int > rat,G: int > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1072433553688619179nt_rat @ H @ S ) @ ( groups1072433553688619179nt_rat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7013_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2555765274223993564er_rat @ H @ S ) @ ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7014_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S: set_int,H: int > nat,G: int > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R @ X12 @ X22 )
              & ( R @ Y12 @ Y22 ) )
           => ( R @ ( times_times_nat @ X12 @ Y12 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1707563613775114915nt_nat @ H @ S ) @ ( groups1707563613775114915nt_nat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_7015_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N5: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N5 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_7016_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N5: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N5 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_7017_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_7018_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7019_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer )
      = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7020_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural )
      = ( groups6325495683096345652atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7021_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7022_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_7023_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_7024_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_7025_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_7026_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_7027_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_7028_sum_Oin__pairs,axiom,
    ! [G: nat > multis2468970476368604999at_nat,M: nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6857163185585827899at_nat
        @ ^ [I4: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7029_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7030_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7031_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7032_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q7: nat] : ( ord_less_nat @ Q7 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_7033_prod_Oin__pairs,axiom,
    ! [G: nat > code_integer,M: nat,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3455450783089532116nteger
        @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7034_prod_Oin__pairs,axiom,
    ! [G: nat > assn,M: nat,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6906906614972039071t_assn
        @ ^ [I4: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7035_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7036_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7037_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7038_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_7039_prod__int__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_7040_sum_Oin__pairs__0,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6857163185585827899at_nat
        @ ^ [I4: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7041_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7042_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7043_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7044_prod_Oin__pairs__0,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3455450783089532116nteger
        @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_7045_prod_Oin__pairs__0,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6906906614972039071t_assn
        @ ^ [I4: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_7046_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_7047_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_7048_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_7049_sum__choose__lower,axiom,
    ! [R3: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_7050_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_7051_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_7052_sum__nonneg__0,axiom,
    ! [S2: set_o,F: $o > code_integer,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups4406642042086082107nteger @ F @ S2 )
            = zero_z3403309356797280102nteger )
         => ( ( member_o @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7053_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: nat > code_integer,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ S2 )
            = zero_z3403309356797280102nteger )
         => ( ( member_nat @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7054_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > code_integer,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ S2 )
            = zero_z3403309356797280102nteger )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7055_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: code_integer > code_integer,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups879477027807139574nteger @ F @ S2 )
            = zero_z3403309356797280102nteger )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7056_sum__nonneg__0,axiom,
    ! [S2: set_o,F: $o > rat,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_o @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7057_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: nat > rat,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_nat @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7058_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > rat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7059_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,F: code_integer > rat,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups6602215022474089585er_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7060_sum__nonneg__0,axiom,
    ! [S2: set_o,F: $o > nat,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ S2 )
            = zero_zero_nat )
         => ( ( member_o @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7061_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > nat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
            = zero_zero_nat )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F @ I2 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7062_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > code_integer,B3: code_integer,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups4406642042086082107nteger @ F @ S2 )
            = B3 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7063_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > code_integer,B3: code_integer,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ S2 )
            = B3 )
         => ( ( member_nat @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7064_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > code_integer,B3: code_integer,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ S2 )
            = B3 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7065_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: code_integer > code_integer,B3: code_integer,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups879477027807139574nteger @ F @ S2 )
            = B3 )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7066_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > rat,B3: rat,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
            = B3 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7067_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > rat,B3: rat,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = B3 )
         => ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7068_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > rat,B3: rat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = B3 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7069_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: code_integer > rat,B3: rat,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups6602215022474089585er_rat @ F @ S2 )
            = B3 )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7070_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > nat,B3: nat,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ S2 )
            = B3 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7071_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > nat,B3: nat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
            = B3 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7072_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_7073_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_7074_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_7075_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_7076_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_7077_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_7078_add__mset__in__multiset,axiom,
    ! [M6: product_prod_int_int > nat,A: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7079_add__mset__in__multiset,axiom,
    ! [M6: list_nat > nat,A: list_nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7080_add__mset__in__multiset,axiom,
    ! [M6: set_nat > nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7081_add__mset__in__multiset,axiom,
    ! [M6: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7082_add__mset__in__multiset,axiom,
    ! [M6: int > nat,A: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7083_add__mset__in__multiset,axiom,
    ! [M6: code_integer > nat,A: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7084_add__mset__in__multiset,axiom,
    ! [M6: product_prod_nat_nat > nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7085_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > code_integer,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups4406642042086082107nteger @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups4406642042086082107nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7086_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > code_integer,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7873554091576472773nteger @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7087_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups879477027807139574nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7088_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > rat,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7089_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > rat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7090_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7091_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > nat,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups8507830703676809646_o_nat @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7092_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > nat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7093_sum_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > nat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( if_nat @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7094_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > int,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups8505340233167759370_o_int @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups8505340233167759370_o_int
          @ ^ [X4: $o] : ( if_int @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ zero_zero_int )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_7095_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: nat > code_integer,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ G @ S ) @ T )
         => ( ( groups7237345082560585321er_nat
              @ ^ [Y4: code_integer] :
                  ( groups3542108847815614940at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7096_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: nat > int,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [Y4: int] :
                  ( groups3542108847815614940at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7097_sum_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: int > nat,H: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ S ) @ T )
         => ( ( groups3539618377306564664at_int
              @ ^ [Y4: nat] :
                  ( groups4538972089207619220nt_int @ H
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups4538972089207619220nt_int @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7098_sum_Ogroup,axiom,
    ! [S: set_int,T: set_Code_integer,G: int > code_integer,H: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ( ord_le7084787975880047091nteger @ ( image_1587234942943678608nteger @ G @ S ) @ T )
         => ( ( groups7234854612051535045er_int
              @ ^ [Y4: code_integer] :
                  ( groups4538972089207619220nt_int @ H
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups4538972089207619220nt_int @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7099_sum_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_o_nat @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y4: nat] :
                  ( groups8507830703676809646_o_nat @ H
                  @ ( collect_o
                    @ ^ [X4: $o] :
                        ( ( member_o @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups8507830703676809646_o_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7100_sum_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: int > nat,H: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y4: nat] :
                  ( groups4541462559716669496nt_nat @ H
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups4541462559716669496nt_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7101_sum_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y4: nat] :
                  ( groups7237345082560585321er_nat @ H
                  @ ( collect_Code_integer
                    @ ^ [X4: code_integer] :
                        ( ( member_Code_integer @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups7237345082560585321er_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7102_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_nat,G: nat > nat,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y4: nat] :
                  ( groups3542108847815614940at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7103_sum_Ogroup,axiom,
    ! [S: set_o,T: set_int,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_int @ T )
       => ( ( ord_less_eq_set_int @ ( image_o_int @ G @ S ) @ T )
         => ( ( groups4538972089207619220nt_int
              @ ^ [Y4: int] :
                  ( groups8505340233167759370_o_int @ H
                  @ ( collect_o
                    @ ^ [X4: $o] :
                        ( ( member_o @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups8505340233167759370_o_int @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7104_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: nat > int,H: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
         => ( ( groups4538972089207619220nt_int
              @ ^ [Y4: int] :
                  ( groups3539618377306564664at_int @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ).

% sum.group
thf(fact_7105_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7873554091576472773nteger @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups7873554091576472773nteger @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7106_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups879477027807139574nteger @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups879477027807139574nteger @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7107_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7108_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7109_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7110_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups7237345082560585321er_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7111_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7234854612051535045er_int @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups7234854612051535045er_int @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7112_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups7501900531339628137nteger @ G
          @ ( minus_minus_set_nat @ A3
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups7501900531339628137nteger @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7113_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( minus_minus_set_nat @ A3
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups2906978787729119204at_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7114_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3539618377306564664at_int @ G
          @ ( minus_minus_set_nat @ A3
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups3539618377306564664at_int @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7115_diff__preserves__multiset,axiom,
    ! [M6: product_prod_int_int > nat,N7: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7116_diff__preserves__multiset,axiom,
    ! [M6: list_nat > nat,N7: list_nat > nat] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7117_diff__preserves__multiset,axiom,
    ! [M6: set_nat > nat,N7: set_nat > nat] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7118_diff__preserves__multiset,axiom,
    ! [M6: nat > nat,N7: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7119_diff__preserves__multiset,axiom,
    ! [M6: int > nat,N7: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7120_diff__preserves__multiset,axiom,
    ! [M6: code_integer > nat,N7: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7121_diff__preserves__multiset,axiom,
    ! [M6: product_prod_nat_nat > nat,N7: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7122_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: nat > code_integer,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ G @ S ) @ T )
         => ( ( groups3190895334310489300er_nat
              @ ^ [Y4: code_integer] :
                  ( groups708209901874060359at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups708209901874060359at_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7123_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_int,G: nat > int,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [Y4: int] :
                  ( groups708209901874060359at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups708209901874060359at_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7124_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_Code_integer,G: nat > code_integer,H: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ G @ S ) @ T )
         => ( ( groups3188404863801439024er_int
              @ ^ [Y4: code_integer] :
                  ( groups705719431365010083at_int @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups705719431365010083at_int @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7125_prod_Ogroup,axiom,
    ! [S: set_int,T: set_Code_integer,G: int > code_integer,H: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ( ord_le7084787975880047091nteger @ ( image_1587234942943678608nteger @ G @ S ) @ T )
         => ( ( groups3188404863801439024er_int
              @ ^ [Y4: code_integer] :
                  ( groups1705073143266064639nt_int @ H
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups1705073143266064639nt_int @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7126_prod_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_o_nat @ G @ S ) @ T )
         => ( ( groups708209901874060359at_nat
              @ ^ [Y4: nat] :
                  ( groups3504817904513533571_o_nat @ H
                  @ ( collect_o
                    @ ^ [X4: $o] :
                        ( ( member_o @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3504817904513533571_o_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7127_prod_Ogroup,axiom,
    ! [S: set_int,T: set_nat,G: int > nat,H: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ S ) @ T )
         => ( ( groups708209901874060359at_nat
              @ ^ [Y4: nat] :
                  ( groups1707563613775114915nt_nat @ H
                  @ ( collect_int
                    @ ^ [X4: int] :
                        ( ( member_int @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups1707563613775114915nt_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7128_prod_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ G @ S ) @ T )
         => ( ( groups708209901874060359at_nat
              @ ^ [Y4: nat] :
                  ( groups3190895334310489300er_nat @ H
                  @ ( collect_Code_integer
                    @ ^ [X4: code_integer] :
                        ( ( member_Code_integer @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3190895334310489300er_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7129_prod_Ogroup,axiom,
    ! [S: set_nat,T: set_nat,G: nat > nat,H: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ S ) @ T )
         => ( ( groups708209901874060359at_nat
              @ ^ [Y4: nat] :
                  ( groups708209901874060359at_nat @ H
                  @ ( collect_nat
                    @ ^ [X4: nat] :
                        ( ( member_nat @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups708209901874060359at_nat @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7130_prod_Ogroup,axiom,
    ! [S: set_o,T: set_nat,G: $o > nat,H: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_o_nat @ G @ S ) @ T )
         => ( ( groups705719431365010083at_int
              @ ^ [Y4: nat] :
                  ( groups3502327434004483295_o_int @ H
                  @ ( collect_o
                    @ ^ [X4: $o] :
                        ( ( member_o @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3502327434004483295_o_int @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7131_prod_Ogroup,axiom,
    ! [S: set_Code_integer,T: set_nat,G: code_integer > nat,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ G @ S ) @ T )
         => ( ( groups705719431365010083at_int
              @ ^ [Y4: nat] :
                  ( groups3188404863801439024er_int @ H
                  @ ( collect_Code_integer
                    @ ^ [X4: code_integer] :
                        ( ( member_Code_integer @ X4 @ S )
                        & ( ( G @ X4 )
                          = Y4 ) ) ) )
              @ T )
            = ( groups3188404863801439024er_int @ H @ S ) ) ) ) ) ).

% prod.group
thf(fact_7132_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_7133_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > code_integer,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7694694392188491536nteger @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups7694694392188491536nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7134_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > code_integer,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups3827104343326376752nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7135_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups3674199335183972705nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7136_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > assn,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups5301882518646026715o_assn @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups5301882518646026715o_assn
          @ ^ [X4: $o] : ( if_assn @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7137_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > assn,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7138_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > assn,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_assn )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7139_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > rat,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups2869687844427037835_o_rat @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups2869687844427037835_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_rat )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7140_prod_Ointer__restrict,axiom,
    ! [A3: set_int,G: int > rat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) )
        = ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( member_int @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_rat )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7141_prod_Ointer__restrict,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
        = ( groups2555765274223993564er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( member_Code_integer @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_rat )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7142_prod_Ointer__restrict,axiom,
    ! [A3: set_o,G: $o > nat,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups3504817904513533571_o_nat @ G @ ( inf_inf_set_o @ A3 @ B3 ) )
        = ( groups3504817904513533571_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( member_o @ X4 @ B3 ) @ ( G @ X4 ) @ one_one_nat )
          @ A3 ) ) ) ).

% prod.inter_restrict
thf(fact_7143_power__numeral__even,axiom,
    ! [Z: code_integer,W2: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W2 ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_7144_power__numeral__even,axiom,
    ! [Z: assn,W2: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_assn @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_7145_power__numeral__even,axiom,
    ! [Z: rat,W2: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_7146_power__numeral__even,axiom,
    ! [Z: nat,W2: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_7147_power__numeral__even,axiom,
    ! [Z: int,W2: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_7148_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3827104343326376752nteger @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7149_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3674199335183972705nteger @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7150_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_assn ) ) ) )
        = ( groups7882442080178216443t_assn @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7151_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_assn ) ) ) )
        = ( groups1304777262505850412r_assn @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7152_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7153_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups2555765274223993564er_rat @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups2555765274223993564er_rat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7154_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7155_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3190895334310489300er_nat @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups3190895334310489300er_nat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7156_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3188404863801439024er_int @ G
          @ ( minus_2355218937544613996nteger @ A3
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups3188404863801439024er_int @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7157_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( minus_minus_set_nat @ A3
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3455450783089532116nteger @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7158_finite__abs__int__segment,axiom,
    ! [A: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [K3: rat] :
            ( ( member_rat @ K3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A ) ) ) ) ).

% finite_abs_int_segment
thf(fact_7159_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_7160_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_7161_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_7162_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_7163_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( ( ord_less_eq_set_assn @ ( set_or7959216805967363635t_assn @ A @ B ) @ ( set_or5523502641901747543n_assn @ C @ D2 ) )
      = ( ( ord_less_eq_assn @ A @ B )
       => ( ( ord_less_eq_assn @ C @ A )
          & ( ord_less_assn @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7164_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D2: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or8585797421378605585et_int @ C @ D2 ) )
      = ( ( ord_less_eq_set_int @ A @ B )
       => ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_set_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7165_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_rat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7166_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
      = ( ( ord_less_eq_num @ A @ B )
       => ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_num @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7167_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
       => ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7168_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
      = ( ( ord_less_eq_int @ A @ B )
       => ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7169_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_7170_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_7171_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7172_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M ) @ ( set_or1222409239386451017an_num @ M @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7173_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7174_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7175_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M )
     => ( ( ord_le6747313008572928689nteger @ M @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7176_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_7177_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_7178_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_7179_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_7180_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_7181_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_7182_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_7183_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_7184_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_7185_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_7186_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_7187_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_7188_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_7189_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_7190_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_3573771949741848930nteger @ ( G @ M ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_7191_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_assn @ ( G @ M ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_7192_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_7193_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_7194_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_7195_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_7196_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_7197_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_7198_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_7199_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_7200_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ N2 ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_7201_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ N2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_7202_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ N2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_7203_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ N2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_7204_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_3573771949741848930nteger @ ( G @ N2 ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_7205_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_assn @ ( G @ N2 ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_7206_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_rat @ ( G @ N2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_7207_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ N2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_7208_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ N2 ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_7209_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups4406642042086082107nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7210_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7211_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7212_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups879477027807139574nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7213_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7214_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7215_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7216_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7217_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,F: $o > nat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7218_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7219_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups879477027807139574nteger @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7220_sum__pos,axiom,
    ! [I5: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups4406642042086082107nteger @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7221_sum__pos,axiom,
    ! [I5: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7222_sum__pos,axiom,
    ! [I5: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7223_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7224_sum__pos,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7225_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7226_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7227_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups7237345082560585321er_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7228_sum__pos,axiom,
    ! [I5: set_o,F: $o > nat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7229_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7230_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7231_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7232_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7233_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7234_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7235_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7236_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7237_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,F: $o > int] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I2 @ I5 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7238_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I2 @ I5 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7239_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7240_less__1__prod,axiom,
    ! [I5: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7241_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7242_less__1__prod,axiom,
    ! [I5: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7243_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7244_less__1__prod,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7245_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7246_less__1__prod,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7247_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7248_less__1__prod,axiom,
    ! [I5: set_o,F: $o > int] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7249_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_7250_sum_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups4406642042086082107nteger @ G @ A3 )
                  = ( groups4406642042086082107nteger @ H @ B3 ) )
                = ( ( groups4406642042086082107nteger @ G @ C3 )
                  = ( groups4406642042086082107nteger @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7251_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups879477027807139574nteger @ G @ A3 )
                  = ( groups879477027807139574nteger @ H @ B3 ) )
                = ( ( groups879477027807139574nteger @ G @ C3 )
                  = ( groups879477027807139574nteger @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7252_sum_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ A3 )
                  = ( groups7872700643590313910_o_rat @ H @ B3 ) )
                = ( ( groups7872700643590313910_o_rat @ G @ C3 )
                  = ( groups7872700643590313910_o_rat @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7253_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ A3 )
                  = ( groups6602215022474089585er_rat @ H @ B3 ) )
                = ( ( groups6602215022474089585er_rat @ G @ C3 )
                  = ( groups6602215022474089585er_rat @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7254_sum_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ A3 )
                  = ( groups8507830703676809646_o_nat @ H @ B3 ) )
                = ( ( groups8507830703676809646_o_nat @ G @ C3 )
                  = ( groups8507830703676809646_o_nat @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7255_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups7237345082560585321er_nat @ G @ A3 )
                  = ( groups7237345082560585321er_nat @ H @ B3 ) )
                = ( ( groups7237345082560585321er_nat @ G @ C3 )
                  = ( groups7237345082560585321er_nat @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7256_sum_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups8505340233167759370_o_int @ G @ A3 )
                  = ( groups8505340233167759370_o_int @ H @ B3 ) )
                = ( ( groups8505340233167759370_o_int @ G @ C3 )
                  = ( groups8505340233167759370_o_int @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7257_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups7234854612051535045er_int @ G @ A3 )
                  = ( groups7234854612051535045er_int @ H @ B3 ) )
                = ( ( groups7234854612051535045er_int @ G @ C3 )
                  = ( groups7234854612051535045er_int @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7258_sum_Osame__carrier,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C3 )
     => ( ( ord_less_eq_set_nat @ A3 @ C3 )
       => ( ( ord_less_eq_set_nat @ B3 @ C3 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups7501900531339628137nteger @ G @ A3 )
                  = ( groups7501900531339628137nteger @ H @ B3 ) )
                = ( ( groups7501900531339628137nteger @ G @ C3 )
                  = ( groups7501900531339628137nteger @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7259_sum_Osame__carrier,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C3 )
     => ( ( ord_less_eq_set_nat @ A3 @ C3 )
       => ( ( ord_less_eq_set_nat @ B3 @ C3 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ A3 )
                  = ( groups2906978787729119204at_rat @ H @ B3 ) )
                = ( ( groups2906978787729119204at_rat @ G @ C3 )
                  = ( groups2906978787729119204at_rat @ H @ C3 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7260_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups4406642042086082107nteger @ G @ C3 )
                  = ( groups4406642042086082107nteger @ H @ C3 ) )
               => ( ( groups4406642042086082107nteger @ G @ A3 )
                  = ( groups4406642042086082107nteger @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7261_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups879477027807139574nteger @ G @ C3 )
                  = ( groups879477027807139574nteger @ H @ C3 ) )
               => ( ( groups879477027807139574nteger @ G @ A3 )
                  = ( groups879477027807139574nteger @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7262_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ C3 )
                  = ( groups7872700643590313910_o_rat @ H @ C3 ) )
               => ( ( groups7872700643590313910_o_rat @ G @ A3 )
                  = ( groups7872700643590313910_o_rat @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7263_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ C3 )
                  = ( groups6602215022474089585er_rat @ H @ C3 ) )
               => ( ( groups6602215022474089585er_rat @ G @ A3 )
                  = ( groups6602215022474089585er_rat @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7264_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ C3 )
                  = ( groups8507830703676809646_o_nat @ H @ C3 ) )
               => ( ( groups8507830703676809646_o_nat @ G @ A3 )
                  = ( groups8507830703676809646_o_nat @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7265_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups7237345082560585321er_nat @ G @ C3 )
                  = ( groups7237345082560585321er_nat @ H @ C3 ) )
               => ( ( groups7237345082560585321er_nat @ G @ A3 )
                  = ( groups7237345082560585321er_nat @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7266_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups8505340233167759370_o_int @ G @ C3 )
                  = ( groups8505340233167759370_o_int @ H @ C3 ) )
               => ( ( groups8505340233167759370_o_int @ G @ A3 )
                  = ( groups8505340233167759370_o_int @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7267_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups7234854612051535045er_int @ G @ C3 )
                  = ( groups7234854612051535045er_int @ H @ C3 ) )
               => ( ( groups7234854612051535045er_int @ G @ A3 )
                  = ( groups7234854612051535045er_int @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7268_sum_Osame__carrierI,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C3 )
     => ( ( ord_less_eq_set_nat @ A3 @ C3 )
       => ( ( ord_less_eq_set_nat @ B3 @ C3 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups7501900531339628137nteger @ G @ C3 )
                  = ( groups7501900531339628137nteger @ H @ C3 ) )
               => ( ( groups7501900531339628137nteger @ G @ A3 )
                  = ( groups7501900531339628137nteger @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7269_sum_Osame__carrierI,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C3 )
     => ( ( ord_less_eq_set_nat @ A3 @ C3 )
       => ( ( ord_less_eq_set_nat @ B3 @ C3 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ C3 )
                  = ( groups2906978787729119204at_rat @ H @ C3 ) )
               => ( ( groups2906978787729119204at_rat @ G @ A3 )
                  = ( groups2906978787729119204at_rat @ H @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7270_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ S )
            = ( groups879477027807139574nteger @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7271_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S )
            = ( groups6602215022474089585er_rat @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7272_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S )
            = ( groups7237345082560585321er_nat @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7273_sum_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ S )
            = ( groups7234854612051535045er_int @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7274_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ S )
            = ( groups7501900531339628137nteger @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7275_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S )
            = ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7276_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S )
            = ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7277_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ S )
            = ( groups7873554091576472773nteger @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7278_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: int > rat] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S )
            = ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7279_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: int > nat] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S )
            = ( groups4541462559716669496nt_nat @ G @ T ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7280_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ T )
            = ( groups879477027807139574nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7281_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T )
            = ( groups6602215022474089585er_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7282_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T )
            = ( groups7237345082560585321er_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7283_sum_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ T )
            = ( groups7234854612051535045er_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7284_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ T )
            = ( groups7501900531339628137nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7285_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T )
            = ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7286_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T )
            = ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7287_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ T )
            = ( groups7873554091576472773nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7288_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: int > rat] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T )
            = ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7289_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: int > nat] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T )
            = ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7290_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups4406642042086082107nteger @ G @ S )
              = ( groups4406642042086082107nteger @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7291_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups879477027807139574nteger @ G @ S )
              = ( groups879477027807139574nteger @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7292_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G @ S )
              = ( groups7872700643590313910_o_rat @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7293_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups6602215022474089585er_rat @ G @ S )
              = ( groups6602215022474089585er_rat @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7294_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ S )
              = ( groups8507830703676809646_o_nat @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7295_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7237345082560585321er_nat @ G @ S )
              = ( groups7237345082560585321er_nat @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7296_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8505340233167759370_o_int @ G @ S )
              = ( groups8505340233167759370_o_int @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7297_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7234854612051535045er_int @ G @ S )
              = ( groups7234854612051535045er_int @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7298_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7501900531339628137nteger @ G @ S )
              = ( groups7501900531339628137nteger @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7299_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S: set_nat,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ S )
              = ( groups2906978787729119204at_rat @ H @ T ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7300_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups4406642042086082107nteger @ G @ T )
              = ( groups4406642042086082107nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7301_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups879477027807139574nteger @ G @ T )
              = ( groups879477027807139574nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7302_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G @ T )
              = ( groups7872700643590313910_o_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7303_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups6602215022474089585er_rat @ G @ T )
              = ( groups6602215022474089585er_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7304_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ T )
              = ( groups8507830703676809646_o_nat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7305_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7237345082560585321er_nat @ G @ T )
              = ( groups7237345082560585321er_nat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7306_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8505340233167759370_o_int @ G @ T )
              = ( groups8505340233167759370_o_int @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7307_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7234854612051535045er_int @ G @ T )
              = ( groups7234854612051535045er_int @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7308_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7501900531339628137nteger @ G @ T )
              = ( groups7501900531339628137nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7309_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ T )
              = ( groups2906978787729119204at_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7310_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups6602215022474089585er_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups6602215022474089585er_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7311_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups7237345082560585321er_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7237345082560585321er_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7312_sum_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups7234854612051535045er_int @ G @ A3 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7234854612051535045er_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7313_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups2906978787729119204at_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7314_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3539618377306564664at_int @ G @ A3 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7315_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups3906332499630173760nt_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7316_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups4541462559716669496nt_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7317_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3542108847815614940at_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7318_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups4538972089207619220nt_int @ G @ A3 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7319_sum_Osubset__diff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( finite6177210948735845034at_nat @ A3 )
       => ( ( groups342789780944988191at_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) @ ( groups342789780944988191at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7320_sum__diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
       => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7321_sum__diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
       => ( ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( groups342789780944988191at_rat @ F @ A3 ) @ ( groups342789780944988191at_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7322_sum__diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
       => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) )
          = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7323_sum__diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
       => ( ( groups975429370522433651at_int @ F @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( groups975429370522433651at_int @ F @ A3 ) @ ( groups975429370522433651at_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7324_sum__diff,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A3 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7325_sum__diff,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A3 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7326_sum__diff,axiom,
    ! [A3: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7327_sum__diff,axiom,
    ! [A3: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A3 @ B3 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7328_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups4406642042086082107nteger @ G @ S )
                = ( groups4406642042086082107nteger @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7329_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7873554091576472773nteger @ G @ S )
                = ( groups7873554091576472773nteger @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7330_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups879477027807139574nteger @ G @ S )
                = ( groups879477027807139574nteger @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7331_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7872700643590313910_o_rat @ G @ S )
                = ( groups7872700643590313910_o_rat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7332_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3906332499630173760nt_rat @ G @ S )
                = ( groups3906332499630173760nt_rat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7333_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups6602215022474089585er_rat @ G @ S )
                = ( groups6602215022474089585er_rat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7334_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_nat ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_nat ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups8507830703676809646_o_nat @ G @ S )
                = ( groups8507830703676809646_o_nat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7335_sum_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_nat ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_nat ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups4541462559716669496nt_nat @ G @ S )
                = ( groups4541462559716669496nt_nat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7336_sum_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_nat ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_nat ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7237345082560585321er_nat @ G @ S )
                = ( groups7237345082560585321er_nat @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7337_sum_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = zero_zero_int ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = zero_zero_int ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups8505340233167759370_o_int @ G @ S )
                = ( groups8505340233167759370_o_int @ H @ T ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_7338_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > code_integer] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups3674199335183972705nteger @ G @ A3 )
          = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups3674199335183972705nteger @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7339_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > code_integer] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3455450783089532116nteger @ G @ A3 )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7340_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > assn] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups1304777262505850412r_assn @ G @ A3 )
          = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups1304777262505850412r_assn @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7341_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups6906906614972039071t_assn @ G @ A3 )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups6906906614972039071t_assn @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7342_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups2555765274223993564er_rat @ G @ A3 )
          = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups2555765274223993564er_rat @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7343_prod_Osubset__diff,axiom,
    ! [B3: set_nat,A3: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups73079841787564623at_rat @ G @ A3 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7344_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups3190895334310489300er_nat @ G @ A3 )
          = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups3190895334310489300er_nat @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7345_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups3188404863801439024er_int @ G @ A3 )
          = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups3188404863801439024er_int @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7346_prod_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > code_integer] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups3827104343326376752nteger @ G @ A3 )
          = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7347_prod_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > assn] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups7882442080178216443t_assn @ G @ A3 )
          = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups7882442080178216443t_assn @ G @ B3 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7348_prod_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_Code_integer ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ A3 )
                  = ( groups7694694392188491536nteger @ H @ B3 ) )
                = ( ( groups7694694392188491536nteger @ G @ C3 )
                  = ( groups7694694392188491536nteger @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7349_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_Code_integer ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ A3 )
                  = ( groups3674199335183972705nteger @ H @ B3 ) )
                = ( ( groups3674199335183972705nteger @ G @ C3 )
                  = ( groups3674199335183972705nteger @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7350_prod_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_assn ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ A3 )
                  = ( groups5301882518646026715o_assn @ H @ B3 ) )
                = ( ( groups5301882518646026715o_assn @ G @ C3 )
                  = ( groups5301882518646026715o_assn @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7351_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_assn ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ A3 )
                  = ( groups1304777262505850412r_assn @ H @ B3 ) )
                = ( ( groups1304777262505850412r_assn @ G @ C3 )
                  = ( groups1304777262505850412r_assn @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7352_prod_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ A3 )
                  = ( groups2869687844427037835_o_rat @ H @ B3 ) )
                = ( ( groups2869687844427037835_o_rat @ G @ C3 )
                  = ( groups2869687844427037835_o_rat @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7353_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ A3 )
                  = ( groups2555765274223993564er_rat @ H @ B3 ) )
                = ( ( groups2555765274223993564er_rat @ G @ C3 )
                  = ( groups2555765274223993564er_rat @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7354_prod_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ A3 )
                  = ( groups3504817904513533571_o_nat @ H @ B3 ) )
                = ( ( groups3504817904513533571_o_nat @ G @ C3 )
                  = ( groups3504817904513533571_o_nat @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7355_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3190895334310489300er_nat @ G @ A3 )
                  = ( groups3190895334310489300er_nat @ H @ B3 ) )
                = ( ( groups3190895334310489300er_nat @ G @ C3 )
                  = ( groups3190895334310489300er_nat @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7356_prod_Osame__carrier,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3502327434004483295_o_int @ G @ A3 )
                  = ( groups3502327434004483295_o_int @ H @ B3 ) )
                = ( ( groups3502327434004483295_o_int @ G @ C3 )
                  = ( groups3502327434004483295_o_int @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7357_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3188404863801439024er_int @ G @ A3 )
                  = ( groups3188404863801439024er_int @ H @ B3 ) )
                = ( ( groups3188404863801439024er_int @ G @ C3 )
                  = ( groups3188404863801439024er_int @ H @ C3 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7358_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_Code_integer ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ C3 )
                  = ( groups7694694392188491536nteger @ H @ C3 ) )
               => ( ( groups7694694392188491536nteger @ G @ A3 )
                  = ( groups7694694392188491536nteger @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7359_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_Code_integer ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ C3 )
                  = ( groups3674199335183972705nteger @ H @ C3 ) )
               => ( ( groups3674199335183972705nteger @ G @ A3 )
                  = ( groups3674199335183972705nteger @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7360_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_assn ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ C3 )
                  = ( groups5301882518646026715o_assn @ H @ C3 ) )
               => ( ( groups5301882518646026715o_assn @ G @ A3 )
                  = ( groups5301882518646026715o_assn @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7361_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_assn ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ C3 )
                  = ( groups1304777262505850412r_assn @ H @ C3 ) )
               => ( ( groups1304777262505850412r_assn @ G @ A3 )
                  = ( groups1304777262505850412r_assn @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7362_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ C3 )
                  = ( groups2869687844427037835_o_rat @ H @ C3 ) )
               => ( ( groups2869687844427037835_o_rat @ G @ A3 )
                  = ( groups2869687844427037835_o_rat @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7363_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ C3 )
                  = ( groups2555765274223993564er_rat @ H @ C3 ) )
               => ( ( groups2555765274223993564er_rat @ G @ A3 )
                  = ( groups2555765274223993564er_rat @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7364_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ C3 )
                  = ( groups3504817904513533571_o_nat @ H @ C3 ) )
               => ( ( groups3504817904513533571_o_nat @ G @ A3 )
                  = ( groups3504817904513533571_o_nat @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7365_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_nat ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3190895334310489300er_nat @ G @ C3 )
                  = ( groups3190895334310489300er_nat @ H @ C3 ) )
               => ( ( groups3190895334310489300er_nat @ G @ A3 )
                  = ( groups3190895334310489300er_nat @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7366_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A3: set_o,B3: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C3 )
     => ( ( ord_less_eq_set_o @ A3 @ C3 )
       => ( ( ord_less_eq_set_o @ B3 @ C3 )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ ( minus_minus_set_o @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o @ B4 @ ( minus_minus_set_o @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3502327434004483295_o_int @ G @ C3 )
                  = ( groups3502327434004483295_o_int @ H @ C3 ) )
               => ( ( groups3502327434004483295_o_int @ G @ A3 )
                  = ( groups3502327434004483295_o_int @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7367_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ C3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ C3 )
       => ( ( ord_le7084787975880047091nteger @ B3 @ C3 )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C3 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_int ) )
           => ( ! [B4: code_integer] :
                  ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ C3 @ B3 ) )
                 => ( ( H @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3188404863801439024er_int @ G @ C3 )
                  = ( groups3188404863801439024er_int @ H @ C3 ) )
               => ( ( groups3188404863801439024er_int @ G @ A3 )
                  = ( groups3188404863801439024er_int @ H @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7368_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ S )
            = ( groups3674199335183972705nteger @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7369_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ S )
            = ( groups1304777262505850412r_assn @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7370_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ S )
            = ( groups2555765274223993564er_rat @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7371_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ S )
            = ( groups3190895334310489300er_nat @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7372_prod_Omono__neutral__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ S )
            = ( groups3188404863801439024er_int @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7373_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ S )
            = ( groups3455450783089532116nteger @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7374_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ S )
            = ( groups6906906614972039071t_assn @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7375_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S )
            = ( groups73079841787564623at_rat @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7376_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ S )
            = ( groups3827104343326376752nteger @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7377_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S: set_int,G: int > assn] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ S )
            = ( groups7882442080178216443t_assn @ G @ T ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7378_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ T )
            = ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7379_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ T )
            = ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7380_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ T )
            = ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7381_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ T )
            = ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7382_prod_Omono__neutral__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ T )
            = ( groups3188404863801439024er_int @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7383_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ T )
            = ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7384_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ T )
            = ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7385_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T )
     => ( ( ord_less_eq_set_nat @ S @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T )
            = ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7386_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ T )
            = ( groups3827104343326376752nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7387_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S: set_int,G: int > assn] :
      ( ( finite_finite_int @ T )
     => ( ( ord_less_eq_set_int @ S @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ T )
            = ( groups7882442080178216443t_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7388_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ S )
              = ( groups7694694392188491536nteger @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7389_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ S )
              = ( groups3674199335183972705nteger @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7390_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > assn,G: $o > assn] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_assn ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ S )
              = ( groups5301882518646026715o_assn @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7391_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1304777262505850412r_assn @ G @ S )
              = ( groups1304777262505850412r_assn @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7392_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ S )
              = ( groups2869687844427037835_o_rat @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7393_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2555765274223993564er_rat @ G @ S )
              = ( groups2555765274223993564er_rat @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7394_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ S )
              = ( groups3504817904513533571_o_nat @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7395_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3190895334310489300er_nat @ G @ S )
              = ( groups3190895334310489300er_nat @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7396_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_o,S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_int ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3502327434004483295_o_int @ G @ S )
              = ( groups3502327434004483295_o_int @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7397_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ X3 )
                = one_one_int ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3188404863801439024er_int @ G @ S )
              = ( groups3188404863801439024er_int @ H @ T ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7398_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ T )
              = ( groups7694694392188491536nteger @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7399_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ T )
              = ( groups3674199335183972705nteger @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7400_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ T )
              = ( groups5301882518646026715o_assn @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7401_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1304777262505850412r_assn @ G @ T )
              = ( groups1304777262505850412r_assn @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7402_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ T )
              = ( groups2869687844427037835_o_rat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7403_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2555765274223993564er_rat @ G @ T )
              = ( groups2555765274223993564er_rat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7404_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ T )
              = ( groups3504817904513533571_o_nat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7405_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3190895334310489300er_nat @ G @ T )
              = ( groups3190895334310489300er_nat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7406_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_o,S: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ T )
     => ( ( ord_less_eq_set_o @ S @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3502327434004483295_o_int @ G @ T )
              = ( groups3502327434004483295_o_int @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7407_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( ord_le7084787975880047091nteger @ S @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3188404863801439024er_int @ G @ T )
              = ( groups3188404863801439024er_int @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7408_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7409_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A3 ) @ ( groups6602215022474089585er_rat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7410_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7411_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A3 ) @ ( groups7237345082560585321er_nat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7412_sum_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A3 ) @ ( groups7234854612051535045er_int @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7413_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A3 ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7414_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A3 ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7415_sum_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7416_sum_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A3 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7417_sum_Ounion__inter,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) )
          = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ A3 ) @ ( groups342789780944988191at_rat @ G @ B3 ) ) ) ) ) ).

% sum.union_inter
thf(fact_7418_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: int > rat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G @ A3 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7419_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat @ G @ A3 )
        = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7420_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: int > nat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G @ A3 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7421_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > nat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat @ G @ A3 )
        = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7422_sum_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > int,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7234854612051535045er_int @ G @ A3 )
        = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7423_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > rat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G @ A3 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7424_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > int,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3539618377306564664at_int @ G @ A3 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7425_sum_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat @ G @ A3 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7426_sum_OInt__Diff,axiom,
    ! [A3: set_int,G: int > int,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4538972089207619220nt_int @ G @ A3 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7427_sum_OInt__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( groups342789780944988191at_rat @ G @ A3 )
        = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_7428_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ A3 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7429_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A3 ) @ ( groups3674199335183972705nteger @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7430_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups3455450783089532116nteger @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A3 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7431_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A3 ) @ ( groups7882442080178216443t_assn @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7432_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A3 ) @ ( groups1304777262505850412r_assn @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7433_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A3 ) @ ( groups6906906614972039071t_assn @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7434_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7435_prod_Ounion__inter,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) )
          = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A3 ) @ ( groups2555765274223993564er_rat @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7436_prod_Ounion__inter,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A3 ) @ ( groups73079841787564623at_rat @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7437_prod_Ounion__inter,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ A3 ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).

% prod.union_inter
thf(fact_7438_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: int > code_integer,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3827104343326376752nteger @ G @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups3827104343326376752nteger @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7439_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > code_integer,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3674199335183972705nteger @ G @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups3674199335183972705nteger @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7440_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > code_integer,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3455450783089532116nteger @ G @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups3455450783089532116nteger @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7441_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: int > assn,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7882442080178216443t_assn @ G @ A3 )
        = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7442_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > assn,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1304777262505850412r_assn @ G @ A3 )
        = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7443_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > assn,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups6906906614972039071t_assn @ G @ A3 )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7444_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: int > rat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat @ G @ A3 )
        = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7445_prod_OInt__Diff,axiom,
    ! [A3: set_Code_integer,G: code_integer > rat,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups2555765274223993564er_rat @ G @ A3 )
        = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7446_prod_OInt__Diff,axiom,
    ! [A3: set_nat,G: nat > rat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups73079841787564623at_rat @ G @ A3 )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7447_prod_OInt__Diff,axiom,
    ! [A3: set_int,G: int > nat,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1707563613775114915nt_nat @ G @ A3 )
        = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_7448_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_Code_integer ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_Code_integer ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7694694392188491536nteger @ G @ S )
                = ( groups7694694392188491536nteger @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7449_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_Code_integer ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_Code_integer ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3827104343326376752nteger @ G @ S )
                = ( groups3827104343326376752nteger @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7450_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_Code_integer ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_Code_integer ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3674199335183972705nteger @ G @ S )
                = ( groups3674199335183972705nteger @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7451_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > assn,G: $o > assn] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_assn ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_assn ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups5301882518646026715o_assn @ G @ S )
                = ( groups5301882518646026715o_assn @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7452_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > assn,G: int > assn] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_assn ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_assn ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7882442080178216443t_assn @ G @ S )
                = ( groups7882442080178216443t_assn @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7453_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_assn ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_assn ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups1304777262505850412r_assn @ G @ S )
                = ( groups1304777262505850412r_assn @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7454_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_rat ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_rat ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups2869687844427037835_o_rat @ G @ S )
                = ( groups2869687844427037835_o_rat @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7455_prod_Omono__neutral__cong,axiom,
    ! [T: set_int,S: set_int,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T )
     => ( ( finite_finite_int @ S )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_rat ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_rat ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups1072433553688619179nt_rat @ G @ S )
                = ( groups1072433553688619179nt_rat @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7456_prod_Omono__neutral__cong,axiom,
    ! [T: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T )
     => ( ( finite6017078050557962740nteger @ S )
       => ( ! [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_rat ) )
         => ( ! [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_rat ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups2555765274223993564er_rat @ G @ S )
                = ( groups2555765274223993564er_rat @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7457_prod_Omono__neutral__cong,axiom,
    ! [T: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T )
     => ( ( finite_finite_o @ S )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( minus_minus_set_o @ T @ S ) )
             => ( ( H @ I3 )
                = one_one_nat ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T ) )
               => ( ( G @ I3 )
                  = one_one_nat ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T ) )
                 => ( ( G @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3504817904513533571_o_nat @ G @ S )
                = ( groups3504817904513533571_o_nat @ H @ T ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
thf(fact_7458_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_7459_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_7460_double__arith__series,axiom,
    ! [A: rat,D2: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_7461_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) )
        @ ( groups6325495683096345652atural
          @ ^ [I4: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I4 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_7462_double__arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_7463_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_7464_double__arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_7465_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_7466_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) ) ).

% double_gauss_sum
thf(fact_7467_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_7468_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum
thf(fact_7469_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_7470_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_7471_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_7472_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_7473_sum__diff__nat,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ A3 )
       => ( ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7474_sum__diff__nat,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7475_sum__diff__nat,axiom,
    ! [B3: set_int,A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7476_sum__diff__nat,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A3 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7477_vandermonde,axiom,
    ! [M: nat,N2: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R3 ) ) ).

% vandermonde
thf(fact_7478_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I4 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_7479_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q6 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_7480_sum_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% sum.nested_swap
thf(fact_7481_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I4 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_7482_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I4 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_7483_prod_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_7484_prod_Onested__swap,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( groups705719431365010083at_int @ ( A @ I4 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_7485_fact__prod__rev,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_7486_fact__prod__rev,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N: nat] : ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_7487_fact__prod__rev,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_7488_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7489_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_7490_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_7491_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_7492_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : X4
          @ ( set_or1266510415728281911st_int @ M @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_7493_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7494_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7495_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7496_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7497_sum_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_int @ ( groups7234854612051535045er_int @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7498_sum_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7499_sum_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7500_sum_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7501_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > int,G: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7502_sum_OIf__cases,axiom,
    ! [A3: set_list_nat,P: list_nat > $o,H: list_nat > rat,G: list_nat > rat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( groups3760926236672600436at_rat
          @ ^ [X4: list_nat] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups3760926236672600436at_rat @ H @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) ) @ ( groups3760926236672600436at_rat @ G @ ( inf_inf_set_list_nat @ A3 @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_7503_prod_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3827104343326376752nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7504_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups3674199335183972705nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7505_prod_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3455450783089532116nteger
          @ ^ [X4: nat] : ( if_Code_integer @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups3455450783089532116nteger @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7506_prod_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > assn,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_assn @ ( groups7882442080178216443t_assn @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7507_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_assn @ ( groups1304777262505850412r_assn @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7508_prod_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > assn,G: nat > assn] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups6906906614972039071t_assn
          @ ^ [X4: nat] : ( if_assn @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7509_prod_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_rat @ ( groups1072433553688619179nt_rat @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7510_prod_OIf__cases,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups2555765274223993564er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_rat @ ( groups2555765274223993564er_rat @ H @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7511_prod_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups73079841787564623at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ H @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7512_prod_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1707563613775114915nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A3 )
        = ( times_times_nat @ ( groups1707563613775114915nt_nat @ H @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7513_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_double
thf(fact_7514_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7515_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7516_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7517_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7518_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7519_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7520_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_7521_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( groups6325495683096345652atural
        @ ^ [I4: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I4 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ D2 ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7522_arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7523_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D2 ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7524_arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7525_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7526_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7527_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7528_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7529_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_7530_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_7531_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_7532_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_7533_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_7534_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_7535_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q6 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_7536_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_7537_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_7538_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_7539_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_7540_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_7541_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A3 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => ( ord_le6747313008572928689nteger @ ( groups3674199335183972705nteger @ F @ A3 ) @ ( groups3674199335183972705nteger @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7542_prod__mono__strict,axiom,
    ! [A3: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ A3 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A3 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_o )
         => ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F @ A3 ) @ ( groups7694694392188491536nteger @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7543_prod__mono__strict,axiom,
    ! [A3: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A3 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F @ A3 ) @ ( groups3455450783089532116nteger @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7544_prod__mono__strict,axiom,
    ! [A3: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A3 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_int )
         => ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F @ A3 ) @ ( groups3827104343326376752nteger @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7545_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A3 ) @ ( groups2555765274223993564er_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7546_prod__mono__strict,axiom,
    ! [A3: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_o )
         => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A3 ) @ ( groups2869687844427037835_o_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7547_prod__mono__strict,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7548_prod__mono__strict,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7549_prod__mono__strict,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A3 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bo3990330152332043303nteger )
         => ( ord_less_nat @ ( groups3190895334310489300er_nat @ F @ A3 ) @ ( groups3190895334310489300er_nat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7550_prod__mono__strict,axiom,
    ! [A3: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A3 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A3 != bot_bot_set_o )
         => ( ord_less_nat @ ( groups3504817904513533571_o_nat @ F @ A3 ) @ ( groups3504817904513533571_o_nat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7551_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B4 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A3 ) @ ( groups4406642042086082107nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7552_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B4 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ A3 ) @ ( groups879477027807139574nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7553_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B4 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A3 ) @ ( groups7501900531339628137nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7554_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7555_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7556_sum__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7557_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > nat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7558_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7559_sum__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > int] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A3 ) @ ( groups8505340233167759370_o_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7560_sum__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7561_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_7562_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( plus_p5714425477246183910nteger @ ( groups7873554091576472773nteger @ G @ A3 ) @ ( groups7873554091576472773nteger @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7563_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_p5714425477246183910nteger @ ( groups879477027807139574nteger @ G @ A3 ) @ ( groups879477027807139574nteger @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7564_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7565_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A3 ) @ ( groups6602215022474089585er_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7566_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7567_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A3 ) @ ( groups7237345082560585321er_nat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7568_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A3 ) @ ( groups7234854612051535045er_int @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7569_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ A3 ) @ ( groups7501900531339628137nteger @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7570_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A3 ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7571_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A3 ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_7572_sum__Un,axiom,
    ! [A3: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7573_sum__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups6602215022474089585er_rat @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ F @ B3 ) ) @ ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7574_sum__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7575_sum__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups7234854612051535045er_int @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ F @ B3 ) ) @ ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7576_sum__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7577_sum__Un,axiom,
    ! [A3: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ A3 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7578_sum__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( groups342789780944988191at_rat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F @ A3 ) @ ( groups342789780944988191at_rat @ F @ B3 ) ) @ ( groups342789780944988191at_rat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7579_sum__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( groups975429370522433651at_int @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups975429370522433651at_int @ F @ A3 ) @ ( groups975429370522433651at_int @ F @ B3 ) ) @ ( groups975429370522433651at_int @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7580_sum__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > rat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B3 )
       => ( ( groups1265588324298845189at_rat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups1265588324298845189at_rat @ F @ A3 ) @ ( groups1265588324298845189at_rat @ F @ B3 ) ) @ ( groups1265588324298845189at_rat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7581_sum__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > int] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B3 )
       => ( ( groups1898227913876290649at_int @ F @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups1898227913876290649at_int @ F @ A3 ) @ ( groups1898227913876290649at_int @ F @ B3 ) ) @ ( groups1898227913876290649at_int @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un
thf(fact_7582_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A3 ) @ ( groups6602215022474089585er_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7583_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A3 ) @ ( groups7237345082560585321er_nat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7584_sum_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A3 ) @ ( groups7234854612051535045er_int @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7585_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A3 ) @ ( groups7872700643590313910_o_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7586_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > nat] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups8507830703676809646_o_nat @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( groups8507830703676809646_o_nat @ G @ A3 ) @ ( groups8507830703676809646_o_nat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7587_sum_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > int] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups8505340233167759370_o_int @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( plus_plus_int @ ( groups8505340233167759370_o_int @ G @ A3 ) @ ( groups8505340233167759370_o_int @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7588_sum_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A3 ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7589_sum_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A3 ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7590_sum_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ( inf_inf_set_int @ A3 @ B3 )
            = bot_bot_set_int )
         => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7591_sum_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ( inf_inf_set_int @ A3 @ B3 )
            = bot_bot_set_int )
         => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_7592_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ A3 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7593_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A3 ) @ ( groups3674199335183972705nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7594_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A3 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7595_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A3 ) @ ( groups7882442080178216443t_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7596_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A3 ) @ ( groups1304777262505850412r_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7597_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A3 ) @ ( groups6906906614972039071t_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7598_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7599_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A3 ) @ ( groups2555765274223993564er_rat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7600_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A3 ) @ ( groups73079841787564623at_rat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7601_prod_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ A3 ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_7602_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A3 ) @ ( groups3674199335183972705nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7603_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > code_integer] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups7694694392188491536nteger @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups7694694392188491536nteger @ G @ A3 ) @ ( groups7694694392188491536nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7604_prod_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A3 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7605_prod_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ( inf_inf_set_int @ A3 @ B3 )
            = bot_bot_set_int )
         => ( ( groups3827104343326376752nteger @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ A3 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7606_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A3 ) @ ( groups1304777262505850412r_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7607_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > assn] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups5301882518646026715o_assn @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups5301882518646026715o_assn @ G @ A3 ) @ ( groups5301882518646026715o_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7608_prod_Ounion__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A3 ) @ ( groups6906906614972039071t_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7609_prod_Ounion__disjoint,axiom,
    ! [A3: set_int,B3: set_int,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ( inf_inf_set_int @ A3 @ B3 )
            = bot_bot_set_int )
         => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A3 ) @ ( groups7882442080178216443t_assn @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7610_prod_Ounion__disjoint,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ( inf_in1364745209274528805nteger @ A3 @ B3 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A3 ) @ ( groups2555765274223993564er_rat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7611_prod_Ounion__disjoint,axiom,
    ! [A3: set_o,B3: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A3 ) @ ( groups2869687844427037835_o_rat @ G @ B3 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_7612_sum__Un2,axiom,
    ! [A3: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B3 ) )
     => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7613_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
     => ( ( groups6602215022474089585er_rat @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7614_sum__Un2,axiom,
    ! [A3: set_int,B3: set_int,F: int > nat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B3 ) )
     => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7615_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
     => ( ( groups7237345082560585321er_nat @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7616_sum__Un2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
     => ( ( groups7234854612051535045er_int @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7617_sum__Un2,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7618_sum__Un2,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7619_sum__Un2,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7620_sum__Un2,axiom,
    ! [A3: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B3 ) )
     => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7621_sum__Un2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
     => ( ( groups342789780944988191at_rat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) @ ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) ) ) @ ( groups342789780944988191at_rat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ).

% sum_Un2
thf(fact_7622_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7623_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7624_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7625_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7626_sum_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7627_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7628_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7629_sum_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7630_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7631_sum_Ounion__diff2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( groups342789780944988191at_rat @ G @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) ) ) @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_7632_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3827104343326376752nteger @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3827104343326376752nteger @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7633_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups3674199335183972705nteger @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7634_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups3455450783089532116nteger @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups3455450783089532116nteger @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7635_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > assn] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7636_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7637_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7638_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7639_prod_Ounion__diff2,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7640_prod_Ounion__diff2,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7641_prod_Ounion__diff2,axiom,
    ! [A3: set_int,B3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ B3 @ A3 ) ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_7642_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_7643_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7644_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7645_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7646_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7647_sum__Un__nat,axiom,
    ! [A3: set_int,B3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7648_sum__Un__nat,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( groups7237345082560585321er_nat @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ F @ B3 ) ) @ ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7649_sum__Un__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B3 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7650_sum__Un__nat,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B3 )
       => ( ( groups1900718384385340925at_nat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups1900718384385340925at_nat @ F @ A3 ) @ ( groups1900718384385340925at_nat @ F @ B3 ) ) @ ( groups1900718384385340925at_nat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7651_sum__Un__nat,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,F: produc4166570645942440679at_nat > nat] :
      ( ( finite1918287321285529104at_nat @ A3 )
     => ( ( finite1918287321285529104at_nat @ B3 )
       => ( ( groups7059897769266147197at_nat @ F @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups7059897769266147197at_nat @ F @ A3 ) @ ( groups7059897769266147197at_nat @ F @ B3 ) ) @ ( groups7059897769266147197at_nat @ F @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7652_sum__Un__nat,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ A3 )
     => ( ( finite4343798906461161616at_nat @ B3 )
       => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A3 ) @ ( groups3860910324918113789at_nat @ F @ B3 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7653_sum__Un__nat,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_7654_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7655_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_7656_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7657_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7658_gchoose__row__sum__weighted,axiom,
    ! [R3: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R3 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R3 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7659_pochhammer__prod,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7660_pochhammer__prod,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7661_pochhammer__prod,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7662_pochhammer__prod,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ I4 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7663_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_7664_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7665_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7666_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_7667_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_7668_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri3624122377584611663nteger @ N2 )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_7669_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_7670_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7671_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B3 @ A3 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups4406642042086082107nteger @ F @ A3 ) @ ( groups4406642042086082107nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7672_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups879477027807139574nteger @ F @ A3 ) @ ( groups879477027807139574nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7673_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A3 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups7501900531339628137nteger @ F @ A3 ) @ ( groups7501900531339628137nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7674_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: $o > rat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7675_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A3 ) @ ( groups6602215022474089585er_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7676_sum__strict__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7677_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: $o > nat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B3 @ A3 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7678_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ ( groups7237345082560585321er_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7679_sum__strict__mono2,axiom,
    ! [B3: set_o,A3: set_o,B: $o,F: $o > int] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B3 @ A3 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A3 ) @ ( groups8505340233167759370_o_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7680_sum__strict__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,B: code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ ( groups7234854612051535045er_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7681_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B4 ) ) )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ A3 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A3 ) @ ( groups7694694392188491536nteger @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7682_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B4 ) ) )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ A3 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A3 ) @ ( groups3674199335183972705nteger @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7683_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B4 ) ) )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ A3 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A3 ) @ ( groups3455450783089532116nteger @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7684_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A3 ) @ ( groups2869687844427037835_o_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7685_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A3 ) @ ( groups2555765274223993564er_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7686_prod__mono2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7687_prod__mono2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > int] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [B4: $o] :
              ( ( member_o @ B4 @ ( minus_minus_set_o @ B3 @ A3 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ A3 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A3 ) @ ( groups3502327434004483295_o_int @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7688_prod__mono2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [B4: code_integer] :
              ( ( member_Code_integer @ B4 @ ( minus_2355218937544613996nteger @ B3 @ A3 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: code_integer] :
                ( ( member_Code_integer @ A4 @ A3 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups3188404863801439024er_int @ F @ A3 ) @ ( groups3188404863801439024er_int @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7689_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A3 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A3 ) @ ( groups3827104343326376752nteger @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7690_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7691_prod__Un,axiom,
    ! [A3: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups1072433553688619179nt_rat @ F @ ( sup_sup_set_int @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ F @ B3 ) ) @ ( groups1072433553688619179nt_rat @ F @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7692_prod__Un,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups2555765274223993564er_rat @ F @ ( sup_su848401254843788991nteger @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ F @ A3 ) @ ( groups2555765274223993564er_rat @ F @ B3 ) ) @ ( groups2555765274223993564er_rat @ F @ ( inf_in1364745209274528805nteger @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7693_prod__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups3442636767675653108at_rat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups3442636767675653108at_rat @ F @ A3 ) @ ( groups3442636767675653108at_rat @ F @ B3 ) ) @ ( groups3442636767675653108at_rat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7694_prod__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > rat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B3 )
       => ( ! [X3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X3 @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups7136053025338371162at_rat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups7136053025338371162at_rat @ F @ A3 ) @ ( groups7136053025338371162at_rat @ F @ B3 ) ) @ ( groups7136053025338371162at_rat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7695_prod__Un,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat,F: produc4166570645942440679at_nat > rat] :
      ( ( finite1918287321285529104at_nat @ A3 )
     => ( ( finite1918287321285529104at_nat @ B3 )
       => ( ! [X3: produc4166570645942440679at_nat] :
              ( ( member6689249552917799696at_nat @ X3 @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups5938585286922990810at_rat @ F @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups5938585286922990810at_rat @ F @ A3 ) @ ( groups5938585286922990810at_rat @ F @ B3 ) ) @ ( groups5938585286922990810at_rat @ F @ ( inf_in1697001100524423349at_nat @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7696_prod__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > rat] :
      ( ( finite4343798906461161616at_nat @ A3 )
     => ( ( finite4343798906461161616at_nat @ B3 )
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups8874911130973611098at_rat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups8874911130973611098at_rat @ F @ A3 ) @ ( groups8874911130973611098at_rat @ F @ B3 ) ) @ ( groups8874911130973611098at_rat @ F @ ( inf_in7913087082777306421at_nat @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7697_prod__Un,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups73079841787564623at_rat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ F @ B3 ) ) @ ( groups73079841787564623at_rat @ F @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_7698_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7699_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_7700_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_7701_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N2 )
      = ( groups7501900531339628137nteger
        @ ^ [K3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ K3 ) ) @ ( power_8256067586552552935nteger @ A @ K3 ) ) @ ( power_8256067586552552935nteger @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_7702_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_7703_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7704_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7705_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N2 )
      = ( groups7501900531339628137nteger
        @ ^ [K3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ K3 ) ) @ ( comm_s8582702949713902594nteger @ A @ K3 ) ) @ ( comm_s8582702949713902594nteger @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7706_gbinomial__altdef__of__nat,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( divide_divide_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K3 @ I4 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) ) ) ) ).

% gbinomial_altdef_of_nat
thf(fact_7707_gbinomial__mult__fact_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( semiri773545260158071498ct_rat @ K ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact'
thf(fact_7708_gbinomial__mult__fact,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( gbinomial_rat @ A @ K ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact
thf(fact_7709_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_7710_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_7711_divmod__step__eq,axiom,
    ! [L: num,R3: nat,Q6: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q6 @ R3 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q6 @ R3 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_7712_divmod__step__eq,axiom,
    ! [L: num,R3: int,Q6: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q6 @ R3 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q6 @ R3 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_7713_divmod__step__eq,axiom,
    ! [L: num,R3: code_integer,Q6: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q6 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q6 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_7714_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7715_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7716_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7717_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7718_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: 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 @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_7719_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_7720_finite__Collect__conjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7721_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7722_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7723_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7724_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7725_finite__Collect__conjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        | ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7726_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7727_finite__Collect__disjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X4: product_prod_int_int] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7728_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7729_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7730_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7731_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7732_finite__Collect__disjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        & ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7733_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7734_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_7735_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_7736_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_7737_finite__Int,axiom,
    ! [F4: set_int,G2: set_int] :
      ( ( ( finite_finite_int @ F4 )
        | ( finite_finite_int @ G2 ) )
     => ( finite_finite_int @ ( inf_inf_set_int @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_7738_finite__Int,axiom,
    ! [F4: set_Code_integer,G2: set_Code_integer] :
      ( ( ( finite6017078050557962740nteger @ F4 )
        | ( finite6017078050557962740nteger @ G2 ) )
     => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_7739_finite__Int,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( ( finite_finite_nat @ F4 )
        | ( finite_finite_nat @ G2 ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_7740_finite__Int,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F4 )
        | ( finite6177210948735845034at_nat @ G2 ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_7741_finite__Un,axiom,
    ! [F4: set_int,G2: set_int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ F4 @ G2 ) )
      = ( ( finite_finite_int @ F4 )
        & ( finite_finite_int @ G2 ) ) ) ).

% finite_Un
thf(fact_7742_finite__Un,axiom,
    ! [F4: set_Code_integer,G2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ F4 @ G2 ) )
      = ( ( finite6017078050557962740nteger @ F4 )
        & ( finite6017078050557962740nteger @ G2 ) ) ) ).

% finite_Un
thf(fact_7743_finite__Un,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F4 @ G2 ) )
      = ( ( finite6177210948735845034at_nat @ F4 )
        & ( finite6177210948735845034at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_7744_finite__Un,axiom,
    ! [F4: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F4 @ G2 ) )
      = ( ( finite4392333629123659920at_nat @ F4 )
        & ( finite4392333629123659920at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_7745_finite__Un,axiom,
    ! [F4: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F4 @ G2 ) )
      = ( ( finite1918287321285529104at_nat @ F4 )
        & ( finite1918287321285529104at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_7746_finite__Un,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F4 @ G2 ) )
      = ( ( finite4343798906461161616at_nat @ F4 )
        & ( finite4343798906461161616at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_7747_finite__Un,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G2 ) )
      = ( ( finite_finite_nat @ F4 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_7748_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_7749_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_7750_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_7751_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7752_finite__Collect__subsets,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B6: set_Code_integer] : ( ord_le7084787975880047091nteger @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7753_finite__Collect__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7754_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7755_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M6 )
     => ? [N5: nat] :
        ! [X7: list_nat] :
          ( ( member_list_nat @ X7 @ M6 )
         => ( ord_less_nat @ ( size_size_list_nat @ X7 ) @ N5 ) ) ) ).

% finite_maxlen
thf(fact_7756_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( ( finite_finite_list_o @ M6 )
     => ? [N5: nat] :
        ! [X7: list_o] :
          ( ( member_list_o @ X7 @ M6 )
         => ( ord_less_nat @ ( size_size_list_o @ X7 ) @ N5 ) ) ) ).

% finite_maxlen
thf(fact_7757_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_7758_not__finite__existsD,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7759_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_1: list_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7760_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7761_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7762_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7763_not__finite__existsD,axiom,
    ! [P: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
     => ? [X_1: code_integer] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7764_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7765_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_nat,R: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7766_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_int,R: $o > int > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7767_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B3: set_Code_integer,R: $o > code_integer > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B3 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7768_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7769_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7770_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_Code_integer,R: nat > code_integer > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7771_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_nat,R: int > nat > $o] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7772_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_int,R: int > int > $o] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7773_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B3: set_Code_integer,R: int > code_integer > $o] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7774_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Code_integer,B3: set_nat,R: code_integer > nat > $o] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite6017078050557962740nteger
                  @ ( collect_Code_integer
                    @ ^ [A5: code_integer] :
                        ( ( member_Code_integer @ A5 @ A3 )
                        & ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7775_finite__has__maximal2,axiom,
    ! [A3: set_o,A: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ A @ A3 )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ( ord_less_eq_o @ A @ X3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7776_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7777_finite__has__maximal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ A @ A3 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7778_finite__has__maximal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A @ A3 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7779_finite__has__maximal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7780_finite__has__maximal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7781_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7782_finite__has__maximal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_7783_finite__has__minimal2,axiom,
    ! [A3: set_o,A: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ A @ A3 )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ( ord_less_eq_o @ X3 @ A )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7784_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7785_finite__has__minimal2,axiom,
    ! [A3: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( member_Code_integer @ A @ A3 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7786_finite__has__minimal2,axiom,
    ! [A3: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A @ A3 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7787_finite__has__minimal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7788_finite__has__minimal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7789_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7790_finite__has__minimal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7791_finite_OemptyI,axiom,
    finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).

% finite.emptyI
thf(fact_7792_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_7793_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_7794_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_7795_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_7796_infinite__imp__nonempty,axiom,
    ! [S: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S )
     => ( S != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
thf(fact_7797_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_7798_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_7799_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_7800_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_7801_all__subset__image,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_set_nat > $o] :
      ( ( ! [B6: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F @ A3 ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A3 )
           => ( P @ ( image_nat_set_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_7802_all__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A3 ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A3 )
           => ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_7803_all__subset__image,axiom,
    ! [F: int > nat,A3: set_int,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A3 ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ A3 )
           => ( P @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_7804_all__subset__image,axiom,
    ! [F: nat > int,A3: set_nat,P: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A3 ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ord_less_eq_set_nat @ B6 @ A3 )
           => ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_7805_all__subset__image,axiom,
    ! [F: int > int,A3: set_int,P: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A3 ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ord_less_eq_set_int @ B6 @ A3 )
           => ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_7806_finite__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_7807_finite__subset,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( finite6017078050557962740nteger @ A3 ) ) ) ).

% finite_subset
thf(fact_7808_finite__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_7809_finite__subset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( finite_finite_int @ B3 )
       => ( finite_finite_int @ A3 ) ) ) ).

% finite_subset
thf(fact_7810_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_7811_infinite__super,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S @ T )
     => ( ~ ( finite6017078050557962740nteger @ S )
       => ~ ( finite6017078050557962740nteger @ T ) ) ) ).

% infinite_super
thf(fact_7812_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ T ) ) ) ).

% infinite_super
thf(fact_7813_infinite__super,axiom,
    ! [S: set_int,T: set_int] :
      ( ( ord_less_eq_set_int @ S @ T )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ T ) ) ) ).

% infinite_super
thf(fact_7814_rev__finite__subset,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_7815_rev__finite__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( finite6017078050557962740nteger @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_7816_rev__finite__subset,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
       => ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_7817_rev__finite__subset,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( finite_finite_int @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_7818_finite__UnI,axiom,
    ! [F4: set_int,G2: set_int] :
      ( ( finite_finite_int @ F4 )
     => ( ( finite_finite_int @ G2 )
       => ( finite_finite_int @ ( sup_sup_set_int @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7819_finite__UnI,axiom,
    ! [F4: set_Code_integer,G2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ F4 )
     => ( ( finite6017078050557962740nteger @ G2 )
       => ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7820_finite__UnI,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( finite6177210948735845034at_nat @ G2 )
       => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7821_finite__UnI,axiom,
    ! [F4: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ F4 )
     => ( ( finite4392333629123659920at_nat @ G2 )
       => ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7822_finite__UnI,axiom,
    ! [F4: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( ( finite1918287321285529104at_nat @ F4 )
     => ( ( finite1918287321285529104at_nat @ G2 )
       => ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7823_finite__UnI,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( finite4343798906461161616at_nat @ G2 )
       => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7824_finite__UnI,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_7825_Un__infinite,axiom,
    ! [S: set_int,T: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7826_Un__infinite,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S )
     => ~ ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7827_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7828_Un__infinite,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ~ ( finite4392333629123659920at_nat @ S )
     => ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7829_Un__infinite,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ~ ( finite1918287321285529104at_nat @ S )
     => ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7830_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7831_Un__infinite,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_7832_infinite__Un,axiom,
    ! [S: set_int,T: set_int] :
      ( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) )
      = ( ~ ( finite_finite_int @ S )
        | ~ ( finite_finite_int @ T ) ) ) ).

% infinite_Un
thf(fact_7833_infinite__Un,axiom,
    ! [S: set_Code_integer,T: set_Code_integer] :
      ( ( ~ ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ S @ T ) ) )
      = ( ~ ( finite6017078050557962740nteger @ S )
        | ~ ( finite6017078050557962740nteger @ T ) ) ) ).

% infinite_Un
thf(fact_7834_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) ) )
      = ( ~ ( finite6177210948735845034at_nat @ S )
        | ~ ( finite6177210948735845034at_nat @ T ) ) ) ).

% infinite_Un
thf(fact_7835_infinite__Un,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T ) ) )
      = ( ~ ( finite4392333629123659920at_nat @ S )
        | ~ ( finite4392333629123659920at_nat @ T ) ) ) ).

% infinite_Un
thf(fact_7836_infinite__Un,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
      ( ( ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) ) )
      = ( ~ ( finite1918287321285529104at_nat @ S )
        | ~ ( finite1918287321285529104at_nat @ T ) ) ) ).

% infinite_Un
thf(fact_7837_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) ) )
      = ( ~ ( finite4343798906461161616at_nat @ S )
        | ~ ( finite4343798906461161616at_nat @ T ) ) ) ).

% infinite_Un
thf(fact_7838_infinite__Un,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_Un
thf(fact_7839_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A8: set_nat] :
            ( ( finite_finite_nat @ A8 )
           => ( ! [B8: set_nat] :
                  ( ( ord_less_set_nat @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_7840_finite__psubset__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [A8: set_int] :
            ( ( finite_finite_int @ A8 )
           => ( ! [B8: set_int] :
                  ( ( ord_less_set_int @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_7841_finite__psubset__induct,axiom,
    ! [A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [A8: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ A8 )
           => ( ! [B8: set_Code_integer] :
                  ( ( ord_le1307284697595431911nteger @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_7842_finite__psubset__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [A8: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A8 )
           => ( ! [B8: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_7843_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ ( image_o_nat @ F @ A3 ) )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ~ ( finite_finite_o
                @ ( collect_o
                  @ ^ [A5: $o] :
                      ( ( member_o @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7844_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_int @ ( image_o_int @ F @ A3 ) )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ~ ( finite_finite_o
                @ ( collect_o
                  @ ^ [A5: $o] :
                      ( ( member_o @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7845_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite6017078050557962740nteger @ ( image_o_Code_integer @ F @ A3 ) )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ~ ( finite_finite_o
                @ ( collect_o
                  @ ^ [A5: $o] :
                      ( ( member_o @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7846_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( member_nat @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7847_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ ( image_nat_int @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( member_nat @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7848_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A3 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( member_nat @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7849_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ ( image_int_nat @ F @ A3 ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A5: int] :
                      ( ( member_int @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7850_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: int > int] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ ( image_int_int @ F @ A3 ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A5: int] :
                      ( ( member_int @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7851_pigeonhole__infinite,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite6017078050557962740nteger @ ( image_1587234942943678608nteger @ F @ A3 ) )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ~ ( finite_finite_int
                @ ( collect_int
                  @ ^ [A5: int] :
                      ( ( member_int @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7852_pigeonhole__infinite,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A3 ) )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ~ ( finite6017078050557962740nteger
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ( ( member_Code_integer @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_7853_nat__seg__image__imp__finite,axiom,
    ! [A3: set_set_nat,F: nat > set_nat,N2: nat] :
      ( ( A3
        = ( image_nat_set_nat @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_7854_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F: nat > nat,N2: nat] :
      ( ( A3
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_7855_nat__seg__image__imp__finite,axiom,
    ! [A3: set_int,F: nat > int,N2: nat] :
      ( ( A3
        = ( image_nat_int @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite_finite_int @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_7856_nat__seg__image__imp__finite,axiom,
    ! [A3: set_Code_integer,F: nat > code_integer,N2: nat] :
      ( ( A3
        = ( image_1215581382706833972nteger @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite6017078050557962740nteger @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_7857_nat__seg__image__imp__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,N2: nat] :
      ( ( A3
        = ( image_5846123807819985514at_nat @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite6177210948735845034at_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_7858_finite__conv__nat__seg__image,axiom,
    ( finite1152437895449049373et_nat
    = ( ^ [A6: set_set_nat] :
        ? [N: nat,F3: nat > set_nat] :
          ( A6
          = ( image_nat_set_nat @ F3
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_7859_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A6: set_nat] :
        ? [N: nat,F3: nat > nat] :
          ( A6
          = ( image_nat_nat @ F3
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_7860_finite__conv__nat__seg__image,axiom,
    ( finite_finite_int
    = ( ^ [A6: set_int] :
        ? [N: nat,F3: nat > int] :
          ( A6
          = ( image_nat_int @ F3
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_7861_finite__conv__nat__seg__image,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A6: set_Code_integer] :
        ? [N: nat,F3: nat > code_integer] :
          ( A6
          = ( image_1215581382706833972nteger @ F3
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_7862_finite__conv__nat__seg__image,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [N: nat,F3: nat > product_prod_nat_nat] :
          ( A6
          = ( image_5846123807819985514at_nat @ F3
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_7863_finite__has__minimal,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7864_finite__has__minimal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7865_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7866_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7867_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7868_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7869_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_7870_finite__has__maximal,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A3 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7871_finite__has__maximal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7872_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7873_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A3 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7874_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A3 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7875_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7876_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_7877_finite__surj,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_7878_finite__surj,axiom,
    ! [A3: set_nat,B3: set_Code_integer,F: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_1215581382706833972nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B3 ) ) ) ).

% finite_surj
thf(fact_7879_finite__surj,axiom,
    ! [A3: set_int,B3: set_nat,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_7880_finite__surj,axiom,
    ! [A3: set_int,B3: set_Code_integer,F: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_1587234942943678608nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B3 ) ) ) ).

% finite_surj
thf(fact_7881_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_nat,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_951025933927791156er_nat @ F @ A3 ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_7882_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_4470545334726330049nteger @ F @ A3 ) )
       => ( finite6017078050557962740nteger @ B3 ) ) ) ).

% finite_surj
thf(fact_7883_finite__surj,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A3 ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_7884_finite__surj,axiom,
    ! [A3: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A3 ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_7885_finite__surj,axiom,
    ! [A3: set_Code_integer,B3: set_int,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_948535463418740880er_int @ F @ A3 ) )
       => ( finite_finite_int @ B3 ) ) ) ).

% finite_surj
thf(fact_7886_finite__surj,axiom,
    ! [A3: set_nat,B3: set_set_nat,F: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
       => ( finite1152437895449049373et_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_7887_finite__subset__image,axiom,
    ! [B3: set_nat,F: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_nat_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7888_finite__subset__image,axiom,
    ! [B3: set_nat,F: code_integer > nat,A3: set_Code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_951025933927791156er_nat @ F @ A3 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A3 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B3
              = ( image_951025933927791156er_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7889_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: nat > code_integer,A3: set_nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_1215581382706833972nteger @ F @ A3 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_1215581382706833972nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7890_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: code_integer > code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_4470545334726330049nteger @ F @ A3 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A3 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B3
              = ( image_4470545334726330049nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7891_finite__subset__image,axiom,
    ! [B3: set_nat,F: int > nat,A3: set_int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_int_nat @ F @ A3 ) )
       => ? [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A3 )
            & ( finite_finite_int @ C5 )
            & ( B3
              = ( image_int_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7892_finite__subset__image,axiom,
    ! [B3: set_Code_integer,F: int > code_integer,A3: set_int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ B3 @ ( image_1587234942943678608nteger @ F @ A3 ) )
       => ? [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A3 )
            & ( finite_finite_int @ C5 )
            & ( B3
              = ( image_1587234942943678608nteger @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7893_finite__subset__image,axiom,
    ! [B3: set_int,F: nat > int,A3: set_nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_nat_int @ F @ A3 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_nat_int @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7894_finite__subset__image,axiom,
    ! [B3: set_int,F: code_integer > int,A3: set_Code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_948535463418740880er_int @ F @ A3 ) )
       => ? [C5: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ C5 @ A3 )
            & ( finite6017078050557962740nteger @ C5 )
            & ( B3
              = ( image_948535463418740880er_int @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7895_finite__subset__image,axiom,
    ! [B3: set_int,F: int > int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ ( image_int_int @ F @ A3 ) )
       => ? [C5: set_int] :
            ( ( ord_less_eq_set_int @ C5 @ A3 )
            & ( finite_finite_int @ C5 )
            & ( B3
              = ( image_int_int @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7896_finite__subset__image,axiom,
    ! [B3: set_set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A3 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_nat_set_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_7897_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A3 )
            & ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7898_ex__finite__subset__image,axiom,
    ! [F: code_integer > nat,A3: set_Code_integer,P: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_951025933927791156er_nat @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A3 )
            & ( P @ ( image_951025933927791156er_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7899_ex__finite__subset__image,axiom,
    ! [F: nat > code_integer,A3: set_nat,P: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_1215581382706833972nteger @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A3 )
            & ( P @ ( image_1215581382706833972nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7900_ex__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_4470545334726330049nteger @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A3 )
            & ( P @ ( image_4470545334726330049nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7901_ex__finite__subset__image,axiom,
    ! [F: int > nat,A3: set_int,P: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ A3 )
            & ( P @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7902_ex__finite__subset__image,axiom,
    ! [F: int > code_integer,A3: set_int,P: set_Code_integer > $o] :
      ( ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ ( image_1587234942943678608nteger @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ A3 )
            & ( P @ ( image_1587234942943678608nteger @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7903_ex__finite__subset__image,axiom,
    ! [F: nat > int,A3: set_nat,P: set_int > $o] :
      ( ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A3 )
            & ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7904_ex__finite__subset__image,axiom,
    ! [F: code_integer > int,A3: set_Code_integer,P: set_int > $o] :
      ( ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ ( image_948535463418740880er_int @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ B6 )
            & ( ord_le7084787975880047091nteger @ B6 @ A3 )
            & ( P @ ( image_948535463418740880er_int @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7905_ex__finite__subset__image,axiom,
    ! [F: int > int,A3: set_int,P: set_int > $o] :
      ( ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_int] :
            ( ( finite_finite_int @ B6 )
            & ( ord_less_eq_set_int @ B6 @ A3 )
            & ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7906_ex__finite__subset__image,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_set_nat > $o] :
      ( ( ? [B6: set_set_nat] :
            ( ( finite1152437895449049373et_nat @ B6 )
            & ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F @ A3 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A3 )
            & ( P @ ( image_nat_set_nat @ F @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_7907_all__finite__subset__image,axiom,
    ! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A3 ) )
           => ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7908_all__finite__subset__image,axiom,
    ! [F: code_integer > nat,A3: set_Code_integer,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_951025933927791156er_nat @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A3 ) )
           => ( P @ ( image_951025933927791156er_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7909_all__finite__subset__image,axiom,
    ! [F: nat > code_integer,A3: set_nat,P: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_1215581382706833972nteger @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A3 ) )
           => ( P @ ( image_1215581382706833972nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7910_all__finite__subset__image,axiom,
    ! [F: code_integer > code_integer,A3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_4470545334726330049nteger @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A3 ) )
           => ( P @ ( image_4470545334726330049nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7911_all__finite__subset__image,axiom,
    ! [F: int > nat,A3: set_int,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ A3 ) )
           => ( P @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7912_all__finite__subset__image,axiom,
    ! [F: int > code_integer,A3: set_int,P: set_Code_integer > $o] :
      ( ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ ( image_1587234942943678608nteger @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ A3 ) )
           => ( P @ ( image_1587234942943678608nteger @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7913_all__finite__subset__image,axiom,
    ! [F: nat > int,A3: set_nat,P: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A3 ) )
           => ( P @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7914_all__finite__subset__image,axiom,
    ! [F: code_integer > int,A3: set_Code_integer,P: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ ( image_948535463418740880er_int @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_Code_integer] :
            ( ( ( finite6017078050557962740nteger @ B6 )
              & ( ord_le7084787975880047091nteger @ B6 @ A3 ) )
           => ( P @ ( image_948535463418740880er_int @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7915_all__finite__subset__image,axiom,
    ! [F: int > int,A3: set_int,P: set_int > $o] :
      ( ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_int] :
            ( ( ( finite_finite_int @ B6 )
              & ( ord_less_eq_set_int @ B6 @ A3 ) )
           => ( P @ ( image_int_int @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7916_all__finite__subset__image,axiom,
    ! [F: nat > set_nat,A3: set_nat,P: set_set_nat > $o] :
      ( ( ! [B6: set_set_nat] :
            ( ( ( finite1152437895449049373et_nat @ B6 )
              & ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F @ A3 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A3 ) )
           => ( P @ ( image_nat_set_nat @ F @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_7917_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_7918_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_7919_set__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_7920_set__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se1617098188084679374atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_7921_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_7922_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7923_unset__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7924_unset__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se7083795435491715335atural @ zero_zero_nat @ A )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7925_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7926_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_7927_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_7928_set__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_7929_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_7930_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_7931_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_7932_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_7933_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_7934_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_7935_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_7936_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_7937_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_7938_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A5: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7939_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A5: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7940_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_7941_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_7942_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_7943_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_7944_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_7945_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_7946_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_7947_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).

% set_bit_greater_eq
thf(fact_7948_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_7949_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_7950_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_7951_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_7952_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_7953_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_7954_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_7955_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_7956_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_7957_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_7958_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7959_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7960_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_7961_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S5: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_7962_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I4 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7963_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I4 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7964_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I4: nat] :
                ( if_Code_integer
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I4 ) )
                @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7965_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I4 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7966_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I4 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7967_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I4: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I4 ) ) @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7968_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X4: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X4 ) ) @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ).

% round_altdef
thf(fact_7969_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S5: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_7970_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_7971_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_7972_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_7973_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_7974_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_7975_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_7976_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_7977_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_7978_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_7979_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_7980_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_7981_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_7982_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_7983_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_7984_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_7985_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_7986_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_7987_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_7988_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_7989_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_7990_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_7991_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_7992_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_7993_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_7994_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_7995_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_7996_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_7997_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_7998_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_7999_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_8000_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_8001_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_8002_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_8003_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_8004_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_8005_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_8006_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_8007_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_8008_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_8009_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_8010_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_8011_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
        = ( divide5121882707175180666atural @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_8012_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_8013_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_8014_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_8015_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_8016_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_8017_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( times_2397367101498566445atural @ A @ B ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_8018_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_8019_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_8020_even__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_8021_even__power,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( power_7079662738309270450atural @ A @ N2 ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_8022_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_8023_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_8024_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_8025_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_8026_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_8027_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_8028_power__less__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_8029_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_8030_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_8031_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_8032_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_8033_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_8034_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_8035_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ one_one_Code_natural )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_8036_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_8037_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_8038_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_8039_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_8040_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_8041_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_8042_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_8043_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_8044_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_8045_even__succ__div__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_8046_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_8047_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_8048_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_8049_even__succ__mod__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_8050_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_8051_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B9: nat,C6: nat] :
          ( ( A
            = ( times_times_nat @ B9 @ C6 ) )
          & ( dvd_dvd_nat @ B9 @ B )
          & ( dvd_dvd_nat @ C6 @ C ) ) ) ).

% division_decomp
thf(fact_8052_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B9: int,C6: int] :
          ( ( A
            = ( times_times_int @ B9 @ C6 ) )
          & ( dvd_dvd_int @ B9 @ B )
          & ( dvd_dvd_int @ C6 @ C ) ) ) ).

% division_decomp
thf(fact_8053_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_8054_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_8055_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).

% dvdE
thf(fact_8056_dvdE,axiom,
    ! [B: assn,A: assn] :
      ( ( dvd_dvd_assn @ B @ A )
     => ~ ! [K2: assn] :
            ( A
           != ( times_times_assn @ B @ K2 ) ) ) ).

% dvdE
thf(fact_8057_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_8058_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_8059_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_8060_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_8061_dvdI,axiom,
    ! [A: assn,B: assn,K: assn] :
      ( ( A
        = ( times_times_assn @ B @ K ) )
     => ( dvd_dvd_assn @ B @ A ) ) ).

% dvdI
thf(fact_8062_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_8063_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_8064_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_8065_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B5: code_integer,A5: code_integer] :
        ? [K3: code_integer] :
          ( A5
          = ( times_3573771949741848930nteger @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_8066_dvd__def,axiom,
    ( dvd_dvd_assn
    = ( ^ [B5: assn,A5: assn] :
        ? [K3: assn] :
          ( A5
          = ( times_times_assn @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_8067_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B5: rat,A5: rat] :
        ? [K3: rat] :
          ( A5
          = ( times_times_rat @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_8068_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B5: nat,A5: nat] :
        ? [K3: nat] :
          ( A5
          = ( times_times_nat @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_8069_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B5: int,A5: int] :
        ? [K3: int] :
          ( A5
          = ( times_times_int @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_8070_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_8071_dvd__mult,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( dvd_dvd_assn @ A @ C )
     => ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% dvd_mult
thf(fact_8072_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_8073_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_8074_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_8075_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_8076_dvd__mult2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ A @ B )
     => ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_8077_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_8078_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_8079_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_8080_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_8081_dvd__mult__left,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
     => ( dvd_dvd_assn @ A @ C ) ) ).

% dvd_mult_left
thf(fact_8082_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_8083_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_8084_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_8085_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_8086_dvd__triv__left,axiom,
    ! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ A @ B ) ) ).

% dvd_triv_left
thf(fact_8087_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_8088_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_8089_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_8090_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D2 )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_8091_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( ( dvd_dvd_assn @ A @ B )
     => ( ( dvd_dvd_assn @ C @ D2 )
       => ( dvd_dvd_assn @ ( times_times_assn @ A @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_8092_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D2 )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_8093_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D2 )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_8094_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D2 )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_8095_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_8096_dvd__mult__right,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
     => ( dvd_dvd_assn @ B @ C ) ) ).

% dvd_mult_right
thf(fact_8097_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_8098_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_8099_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_8100_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_8101_dvd__triv__right,axiom,
    ! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ A ) ) ).

% dvd_triv_right
thf(fact_8102_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_8103_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_8104_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_8105_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_8106_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8107_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8108_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8109_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8110_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8111_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8112_pinf_I9_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X7 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_8113_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_8114_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_8115_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_8116_pinf_I10_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X7 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_8117_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ Z2 @ X7 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_8118_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z2 @ X7 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_8119_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ Z2 @ X7 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_8120_minf_I9_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z2 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_8121_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_8122_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_8123_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_8124_minf_I10_J,axiom,
    ! [D2: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X7: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X7 @ Z2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_8125_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X7: rat] :
      ( ( ord_less_rat @ X7 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_8126_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X7 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_8127_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X7: int] :
      ( ( ord_less_int @ X7 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_8128_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_8129_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_8130_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_8131_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_8132_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_8133_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_8134_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_8135_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_8136_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_8137_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_8138_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_8139_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_8140_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_8141_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_8142_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_8143_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_8144_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_8145_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_8146_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_8147_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_8148_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_8149_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D2 @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D2 ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_8150_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D2 @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D2 ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_8151_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D2 @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D2 ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_8152_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ A )
     => ( ( dvd_dvd_Code_natural @ D2 @ C )
       => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ ( divide5121882707175180666atural @ C @ D2 ) )
          = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_8153_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_8154_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_8155_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_8156_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B )
     => ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_8157_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_8158_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_8159_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_8160_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ C ) @ A )
     => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_8161_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_8162_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_8163_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_8164_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_8165_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_8166_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_8167_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_8168_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_8169_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_8170_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_8171_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_8172_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ C ) @ A )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_8173_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_8174_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_8175_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_8176_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_8177_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_8178_power__le__dvd,axiom,
    ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_8179_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_8180_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_8181_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_8182_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_8183_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_8184_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_8185_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_8186_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_8187_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_8188_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_8189_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).

% fact_dvd
thf(fact_8190_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_8191_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X4: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
      = ( ? [X4: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_8192_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
      = ( ? [X4: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_8193_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_8194_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_8195_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_8196_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_8197_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_8198_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,T5: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X7: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ T5 ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T5 ) ) ) ) ) ).

% inf_period(4)
thf(fact_8199_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,T5: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X7: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ T5 ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) @ T5 ) ) ) ) ) ).

% inf_period(4)
thf(fact_8200_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,T5: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X7: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T5 ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) @ T5 ) ) ) ) ) ).

% inf_period(4)
thf(fact_8201_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,T5: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X7: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X7 @ T5 ) )
          = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X7 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T5 ) ) ) ) ).

% inf_period(3)
thf(fact_8202_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,T5: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X7: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X7 @ T5 ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X7 @ ( times_times_rat @ K4 @ D3 ) ) @ T5 ) ) ) ) ).

% inf_period(3)
thf(fact_8203_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,T5: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X7: int,K4: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X7 @ T5 ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X7 @ ( times_times_int @ K4 @ D3 ) ) @ T5 ) ) ) ) ).

% inf_period(3)
thf(fact_8204_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D2 )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D2 @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_8205_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D2 )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D2 @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_8206_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D2 )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D2 @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_8207_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( ( dvd_dvd_Code_natural @ A @ B )
         => ( ( dvd_dvd_Code_natural @ C @ D2 )
           => ( ( ( divide5121882707175180666atural @ B @ A )
                = ( divide5121882707175180666atural @ D2 @ C ) )
              = ( ( times_2397367101498566445atural @ B @ C )
                = ( times_2397367101498566445atural @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_8208_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_8209_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_8210_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_8211_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_8212_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_8213_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_8214_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_8215_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
          = ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_8216_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_8217_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_8218_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_8219_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ A @ B )
       => ( ( ( divide5121882707175180666atural @ B @ A )
            = C )
          = ( B
            = ( times_2397367101498566445atural @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_8220_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_8221_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_8222_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_8223_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_8224_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_8225_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_8226_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_8227_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_8228_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_8229_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_8230_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_8231_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_8232_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_8233_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_8234_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_8235_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_8236_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_8237_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_8238_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_8239_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( A
          = ( divide5121882707175180666atural @ C @ B ) )
        = ( ( times_2397367101498566445atural @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_8240_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_8241_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_8242_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_8243_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = C )
        = ( A
          = ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_8244_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_8245_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A3 ) @ ( groups3190895334310489300er_nat @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8246_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A3 ) @ ( groups3188404863801439024er_int @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8247_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A3 ) @ ( groups3455450783089532116nteger @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8248_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( dvd_dvd_Code_integer @ ( groups3674199335183972705nteger @ F @ A3 ) @ ( groups3674199335183972705nteger @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8249_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) @ ( groups1707563613775114915nt_nat @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8250_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A3 ) @ ( groups3827104343326376752nteger @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8251_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ ( groups708209901874060359at_nat @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8252_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A3 ) @ ( groups705719431365010083at_int @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8253_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A3: set_int,F: int > int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A3 ) @ ( groups1705073143266064639nt_int @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8254_prod__dvd__prod__subset,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
       => ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A3 ) @ ( groups4077766827762148844at_nat @ F @ B3 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8255_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups3504817904513533571_o_nat @ F @ A3 ) @ ( groups3504817904513533571_o_nat @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8256_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [A4: code_integer] :
              ( ( member_Code_integer @ A4 @ A3 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A3 ) @ ( groups3190895334310489300er_nat @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8257_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups3502327434004483295_o_int @ F @ A3 ) @ ( groups3502327434004483295_o_int @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8258_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [A4: code_integer] :
              ( ( member_Code_integer @ A4 @ A3 )
             => ( dvd_dvd_int @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A3 ) @ ( groups3188404863801439024er_int @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8259_prod__dvd__prod__subset2,axiom,
    ! [B3: set_o,A3: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ B3 )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( dvd_dvd_Code_integer @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups7694694392188491536nteger @ F @ A3 ) @ ( groups7694694392188491536nteger @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8260_prod__dvd__prod__subset2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
             => ( dvd_dvd_Code_integer @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A3 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8261_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ! [A4: code_integer] :
              ( ( member_Code_integer @ A4 @ A3 )
             => ( dvd_dvd_Code_integer @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3674199335183972705nteger @ F @ A3 ) @ ( groups3674199335183972705nteger @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8262_prod__dvd__prod__subset2,axiom,
    ! [B3: set_int,A3: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ A3 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8263_prod__dvd__prod__subset2,axiom,
    ! [B3: set_int,A3: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ A3 )
             => ( dvd_dvd_Code_integer @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A3 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8264_prod__dvd__prod__subset2,axiom,
    ! [B3: set_nat,A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ! [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
             => ( dvd_dvd_nat @ ( F @ A4 ) @ ( G @ A4 ) ) )
         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A3 ) @ ( groups708209901874060359at_nat @ G @ B3 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8265_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_8266_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_8267_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_8268_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8269_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8270_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8271_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8272_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_8273_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q6: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q6 )
          = ( modulo_modulo_nat @ N2 @ Q6 ) )
        = ( dvd_dvd_nat @ Q6 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_8274_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_8275_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_8276_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_8277_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M3: nat,N: nat] :
        ( ( dvd_dvd_nat @ M3 @ N )
        & ( M3 != N ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_8278_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_8279_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_8280_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_8281_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ A ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_8282_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_8283_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_8284_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_8285_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ A @ B ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_8286_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B4 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B4 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_8287_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( ( dvd_dvd_int @ B4 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B4 )
                 => ( ( ( divide_divide_int @ one_one_int @ B4 )
                      = A )
                   => ( ( ( times_times_int @ A @ B4 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_8288_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B4 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B4 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_8289_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B4: code_natural] :
              ( ( B4 != zero_z2226904508553997617atural )
             => ( ( dvd_dvd_Code_natural @ B4 @ one_one_Code_natural )
               => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A )
                    = B4 )
                 => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B4 )
                      = A )
                   => ( ( ( times_2397367101498566445atural @ A @ B4 )
                        = one_one_Code_natural )
                     => ( ( divide5121882707175180666atural @ C @ A )
                       != ( times_2397367101498566445atural @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_8290_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_8291_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_8292_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_8293_evenE,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_natural] :
            ( A
           != ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B4 ) ) ) ).

% evenE
thf(fact_8294_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N2: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_8295_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_8296_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_8297_dvd__power,axiom,
    ! [N2: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_8298_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_8299_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_8300_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_8301_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_8302_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_8303_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% choose_dvd
thf(fact_8304_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% choose_dvd
thf(fact_8305_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_8306_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_8307_dvd__minus__add,axiom,
    ! [Q6: nat,N2: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q6 @ N2 )
     => ( ( ord_less_eq_nat @ Q6 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q6 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q6 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_8308_mod__nat__eqI,axiom,
    ! [R3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N2 )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_8309_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_8310_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_8311_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_8312_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_8313_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_8314_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_8315_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_8316_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_8317_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_8318_sum__div__partition,axiom,
    ! [A3: set_int,F: int > nat,B: nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( divide_divide_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ B )
        = ( plus_plus_nat
          @ ( groups4541462559716669496nt_nat
            @ ^ [A5: int] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A3
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups4541462559716669496nt_nat @ F
              @ ( inf_inf_set_int @ A3
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8319_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat,B: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( divide_divide_nat @ ( groups7237345082560585321er_nat @ F @ A3 ) @ B )
        = ( plus_plus_nat
          @ ( groups7237345082560585321er_nat
            @ ^ [A5: code_integer] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A3
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups7237345082560585321er_nat @ F
              @ ( inf_in1364745209274528805nteger @ A3
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8320_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: code_integer > int,B: int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( divide_divide_int @ ( groups7234854612051535045er_int @ F @ A3 ) @ B )
        = ( plus_plus_int
          @ ( groups7234854612051535045er_int
            @ ^ [A5: code_integer] : ( divide_divide_int @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A3
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups7234854612051535045er_int @ F
              @ ( inf_in1364745209274528805nteger @ A3
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8321_sum__div__partition,axiom,
    ! [A3: set_nat,F: nat > int,B: int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( divide_divide_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ B )
        = ( plus_plus_int
          @ ( groups3539618377306564664at_int
            @ ^ [A5: nat] : ( divide_divide_int @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A3
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups3539618377306564664at_int @ F
              @ ( inf_inf_set_nat @ A3
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8322_sum__div__partition,axiom,
    ! [A3: set_int,F: int > code_integer,B: code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( divide6298287555418463151nteger @ ( groups7873554091576472773nteger @ F @ A3 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7873554091576472773nteger
            @ ^ [A5: int] : ( divide6298287555418463151nteger @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A3
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7873554091576472773nteger @ F
              @ ( inf_inf_set_int @ A3
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8323_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_integer,B: code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( divide6298287555418463151nteger @ ( groups879477027807139574nteger @ F @ A3 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups879477027807139574nteger
            @ ^ [A5: code_integer] : ( divide6298287555418463151nteger @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A3
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups879477027807139574nteger @ F
              @ ( inf_in1364745209274528805nteger @ A3
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8324_sum__div__partition,axiom,
    ! [A3: set_nat,F: nat > code_integer,B: code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( divide6298287555418463151nteger @ ( groups7501900531339628137nteger @ F @ A3 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7501900531339628137nteger
            @ ^ [A5: nat] : ( divide6298287555418463151nteger @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A3
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7501900531339628137nteger @ F
              @ ( inf_inf_set_nat @ A3
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8325_sum__div__partition,axiom,
    ! [A3: set_int,F: int > code_natural,B: code_natural] :
      ( ( finite_finite_int @ A3 )
     => ( ( divide5121882707175180666atural @ ( groups6697149243333190288atural @ F @ A3 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups6697149243333190288atural
            @ ^ [A5: int] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A3
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups6697149243333190288atural @ F
              @ ( inf_inf_set_int @ A3
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8326_sum__div__partition,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_natural,B: code_natural] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( divide5121882707175180666atural @ ( groups8926444216418632897atural @ F @ A3 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups8926444216418632897atural
            @ ^ [A5: code_integer] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A3
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups8926444216418632897atural @ F
              @ ( inf_in1364745209274528805nteger @ A3
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8327_sum__div__partition,axiom,
    ! [A3: set_nat,F: nat > code_natural,B: code_natural] :
      ( ( finite_finite_nat @ A3 )
     => ( ( divide5121882707175180666atural @ ( groups6325495683096345652atural @ F @ A3 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups6325495683096345652atural
            @ ^ [A5: nat] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A3
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups6325495683096345652atural @ F
              @ ( inf_inf_set_nat @ A3
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8328_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_8329_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_8330_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_8331_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_8332_oddE,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_natural] :
            ( A
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_8333_zero__le__even__power,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_8334_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_8335_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_8336_zero__le__odd__power,axiom,
    ! [N2: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ).

% zero_le_odd_power
thf(fact_8337_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_8338_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_8339_zero__le__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_8340_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_8341_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_8342_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_8343_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_8344_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_8345_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ! [X3: set_nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: set_nat] :
                ( ( member_set_nat @ X3 @ A3 )
               => ? [Y6: set_nat] :
                    ( ( member_set_nat @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8346_finite__transitivity__chain,axiom,
    ! [A3: set_Code_integer,R: code_integer > code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ! [X3: code_integer] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: code_integer,Y3: code_integer,Z2: code_integer] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ A3 )
               => ? [Y6: code_integer] :
                    ( ( member_Code_integer @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8347_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [X3: product_prod_nat_nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A3 )
               => ? [Y6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8348_finite__transitivity__chain,axiom,
    ! [A3: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X3: $o] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: $o,Y3: $o,Z2: $o] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ? [Y6: $o] :
                    ( ( member_o @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8349_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X3: nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
               => ? [Y6: nat] :
                    ( ( member_nat @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8350_finite__transitivity__chain,axiom,
    ! [A3: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X3: int] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ A3 )
               => ? [Y6: int] :
                    ( ( member_int @ Y6 @ A3 )
                    & ( R @ X3 @ Y6 ) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8351_zero__less__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_8352_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_8353_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_8354_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( ( ord_less_nat @ K @ M5 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M5 @ N9 )
              & ( member_nat @ N9 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_8355_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M3: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( member_nat @ N @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_8356_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M3: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( member_nat @ N @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_8357_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_8358_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_8359_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_8360_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_8361_power__le__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_8362_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_8363_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_8364_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_8365_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_8366_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_8367_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 )
          = zero_z2226904508553997617atural )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_8368_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_8369_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_8370_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_8371_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 )
          = zero_z2226904508553997617atural )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_8372_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_8373_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_8374_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_8375_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M3: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_8376_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M3: int] :
          ? [N: int] :
            ( ( ord_less_int @ M3 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_8377_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_8378_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_8379_in__finite__psubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B3 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A3 @ B3 )
        & ( finite_finite_nat @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_8380_in__finite__psubset,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A3 @ B3 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A3 @ B3 )
        & ( finite_finite_int @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_8381_in__finite__psubset,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A3 @ B3 ) @ finite2416775604798480986nteger )
      = ( ( ord_le1307284697595431911nteger @ A3 @ B3 )
        & ( finite6017078050557962740nteger @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_8382_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B3 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A3 @ B3 )
        & ( finite6177210948735845034at_nat @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_8383_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8384_flip__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se168947363167071951atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8385_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8386_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8387_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X4: nat] :
          ( collect_nat
          @ ^ [N: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% set_decode_def
thf(fact_8388_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N: nat,A5: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_8389_take__bit__rec,axiom,
    ( bit_se569199155075624693atural
    = ( ^ [N: nat,A5: code_natural] : ( if_Code_natural @ ( N = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_8390_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A5: nat] : ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_8391_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A5: int] : ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_8392_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q7: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_8393_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q7: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_8394_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q7: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_8395_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_8396_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_8397_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_8398_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_8399_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8400_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8401_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8402_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8403_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8404_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8405_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_8406_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_8407_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_8408_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_o @ ( F @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8409_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8410_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > int] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_int @ ( F @ A @ B ) @ ( image_2483585944268220136at_int @ ( produc6840382203811409530at_int @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8411_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
     => ( member_o @ ( F @ A @ B ) @ ( image_2135063354759101220_int_o @ ( produc4947309494688390418_int_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8412_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
     => ( member_int @ ( F @ A @ B ) @ ( image_5042161079198086560nt_int @ ( produc8211389475949308722nt_int @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8413_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_set_nat @ ( F @ A @ B ) @ ( image_15824709712370754et_nat @ ( produc6189476227299908564et_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8414_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8415_pair__imageI,axiom,
    ! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
     => ( member5262025264175285858nt_int @ ( F @ A @ B ) @ ( image_2653370878348428101nt_int @ ( produc4245557441103728435nt_int @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8416_pair__imageI,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A3: set_Pr7098220151150636591it_nat,F: a > produc6653097349344004940it_nat > $o] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ A3 )
     => ( member_o @ ( F @ A @ B ) @ ( image_1991556864075471768_nat_o @ ( produc8860486935419594360_nat_o @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8417_pair__imageI,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A3: set_Pr7098220151150636591it_nat,F: a > produc6653097349344004940it_nat > nat] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ A3 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_1030127162643715024at_nat @ ( produc7727125748877983984at_nat @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_8418_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8419_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8420_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8421_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8422_case__prod__conv,axiom,
    ! [F: int > int > int,A: int,B: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8423_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc3610074571335452306at_nat @ ( produc27273713700761075at_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_8424_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o] :
      ( ( produc3704529784387675049_nat_o @ ( produc8739625826339149834_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_8425_curry__case__prod,axiom,
    ! [F: int > int > product_prod_int_int] :
      ( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_8426_curry__case__prod,axiom,
    ! [F: int > int > $o] :
      ( ( produc175634133007206835_int_o @ ( produc4947309494688390418_int_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_8427_curry__case__prod,axiom,
    ! [F: int > int > int] :
      ( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_8428_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat @ ( produc3610074571335452306at_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_8429_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o @ ( produc3704529784387675049_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_8430_case__prod__curry,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_8431_case__prod__curry,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o @ ( produc175634133007206835_int_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_8432_case__prod__curry,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_8433_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_8434_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_8435_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_8436_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_8437_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_8438_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_8439_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_8440_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_8441_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_8442_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_8443_sgn__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_mult_self_eq
thf(fact_8444_sgn__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_mult_self_eq
thf(fact_8445_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_8446_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_8447_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_8448_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_8449_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_8450_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_8451_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_8452_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: int > code_integer,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups7873554091576472773nteger @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8453_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups879477027807139574nteger @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8454_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: nat > code_integer,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups7501900531339628137nteger @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8455_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8456_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: code_integer > rat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8457_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8458_sum__mult__of__bool__eq,axiom,
    ! [A3: set_int,F: int > nat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( F @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8459_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( times_times_nat @ ( F @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8460_sum__mult__of__bool__eq,axiom,
    ! [A3: set_Code_integer,F: code_integer > int,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( times_times_int @ ( F @ X4 ) @ ( zero_n2684676970156552555ol_int @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8461_sum__mult__of__bool__eq,axiom,
    ! [A3: set_nat,F: nat > int,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( times_times_int @ ( F @ X4 ) @ ( zero_n2684676970156552555ol_int @ ( P @ X4 ) ) )
          @ A3 )
        = ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8462_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,P: int > $o,F: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups7873554091576472773nteger @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8463_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups879477027807139574nteger @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8464_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,P: nat > $o,F: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups7501900531339628137nteger @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8465_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,P: int > $o,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8466_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8467_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,P: nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8468_sum__of__bool__mult__eq,axiom,
    ! [A3: set_int,P: int > $o,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8469_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8470_sum__of__bool__mult__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8471_sum__of__bool__mult__eq,axiom,
    ! [A3: set_nat,P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P @ X4 ) ) @ ( F @ X4 ) )
          @ A3 )
        = ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8472_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N2 @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_8473_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se569199155075624693atural @ M @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ N2 @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_8474_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_8475_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_8476_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_8477_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se569199155075624693atural @ N2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_8478_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_8479_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_8480_nested__case__prod__simp,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F3: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( produc2626176000494625587at_nat
          @ ^ [A5: nat,B5: nat] : ( F3 @ A5 @ B5 @ Y4 )
          @ X4 ) ) ) ).

% nested_case_prod_simp
thf(fact_8481_nested__case__prod__simp,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F3: nat > nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [A5: nat,B5: nat] : ( F3 @ A5 @ B5 @ Y4 )
          @ X4 ) ) ) ).

% nested_case_prod_simp
thf(fact_8482_case__prod__app,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F3: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( produc2626176000494625587at_nat
          @ ^ [L2: nat,R4: nat] : ( F3 @ L2 @ R4 @ Y4 )
          @ X4 ) ) ) ).

% case_prod_app
thf(fact_8483_case__prod__app,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F3: nat > nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [L2: nat,R4: nat] : ( F3 @ L2 @ R4 @ Y4 )
          @ X4 ) ) ) ).

% case_prod_app
thf(fact_8484_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8485_prod_Ocase__distrib,axiom,
    ! [H: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8486_prod_Ocase__distrib,axiom,
    ! [H: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8487_prod_Ocase__distrib,axiom,
    ! [H: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8488_prod_Ocase__distrib,axiom,
    ! [H: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8489_prod_Ocase__distrib,axiom,
    ! [H: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8490_prod_Ocase__distrib,axiom,
    ! [H: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8491_prod_Ocase__distrib,axiom,
    ! [H: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8492_prod_Ocase__distrib,axiom,
    ! [H: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X13: int,X23: int] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8493_prod_Ocase__distrib,axiom,
    ! [H: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X13: nat,X23: nat] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_8494_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_8495_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_8496_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q6 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q6 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_8497_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_8498_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_8499_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_8500_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_8501_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_8502_split__cong,axiom,
    ! [Q6: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P3: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q6 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P3 = Q6 )
       => ( ( produc27273713700761075at_nat @ F @ P3 )
          = ( produc27273713700761075at_nat @ G @ Q6 ) ) ) ) ).

% split_cong
thf(fact_8503_split__cong,axiom,
    ! [Q6: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q6 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P3 = Q6 )
       => ( ( produc8739625826339149834_nat_o @ F @ P3 )
          = ( produc8739625826339149834_nat_o @ G @ Q6 ) ) ) ) ).

% split_cong
thf(fact_8504_split__cong,axiom,
    ! [Q6: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P3: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( ( product_Pair_int_int @ X3 @ Y3 )
            = Q6 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P3 = Q6 )
       => ( ( produc4245557441103728435nt_int @ F @ P3 )
          = ( produc4245557441103728435nt_int @ G @ Q6 ) ) ) ) ).

% split_cong
thf(fact_8505_split__cong,axiom,
    ! [Q6: product_prod_int_int,F: int > int > $o,G: int > int > $o,P3: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( ( product_Pair_int_int @ X3 @ Y3 )
            = Q6 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P3 = Q6 )
       => ( ( produc4947309494688390418_int_o @ F @ P3 )
          = ( produc4947309494688390418_int_o @ G @ Q6 ) ) ) ) ).

% split_cong
thf(fact_8506_split__cong,axiom,
    ! [Q6: product_prod_int_int,F: int > int > int,G: int > int > int,P3: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( ( product_Pair_int_int @ X3 @ Y3 )
            = Q6 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P3 = Q6 )
       => ( ( produc8211389475949308722nt_int @ F @ P3 )
          = ( produc8211389475949308722nt_int @ G @ Q6 ) ) ) ) ).

% split_cong
thf(fact_8507_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_8508_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_8509_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_8510_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X2: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_8511_old_Oprod_Ocase,axiom,
    ! [F: int > int > int,X1: int,X2: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_8512_case__prod__Pair__iden,axiom,
    ! [P3: produc3925858234332021118et_nat] :
      ( ( produc4058941399401191971et_nat @ produc5001842942810119800et_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8513_case__prod__Pair__iden,axiom,
    ! [P3: produc2732055786443039994et_nat] :
      ( ( produc2377985495875741467et_nat @ produc2245416461498447860et_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8514_case__prod__Pair__iden,axiom,
    ! [P3: produc2285326912895808259nt_int] :
      ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8515_case__prod__Pair__iden,axiom,
    ! [P3: produc7773217078559923341nt_int] :
      ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8516_case__prod__Pair__iden,axiom,
    ! [P3: produc3260487557148687353it_nat] :
      ( ( produc2183267410226956569it_nat @ produc9178034014595674355it_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8517_case__prod__Pair__iden,axiom,
    ! [P3: product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_8518_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_8519_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_8520_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_8521_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_8522_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_8523_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_8524_cond__case__prod__eta,axiom,
    ! [F: int > int > int,G: product_prod_int_int > int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc8211389475949308722nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_8525_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_8526_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_8527_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X4: int,Y4: int] : ( F @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_8528_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X4: int,Y4: int] : ( F @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_8529_case__prod__eta,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int
        @ ^ [X4: int,Y4: int] : ( F @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_8530_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_8531_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_8532_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_8533_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_8534_case__prodE2,axiom,
    ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
      ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_8535_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_8536_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_8537_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_8538_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_8539_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_8540_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_8541_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_8542_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_8543_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_8544_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_8545_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_8546_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_8547_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_8548_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_8549_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_8550_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_8551_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_8552_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_8553_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_8554_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_8555_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_8556_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_8557_uncurry__def,axiom,
    uncurr8011562610307062878at_nat = produc27273713700761075at_nat ).

% uncurry_def
thf(fact_8558_uncurry__def,axiom,
    uncurr7511940902602773877_nat_o = produc8739625826339149834_nat_o ).

% uncurry_def
thf(fact_8559_uncurry__def,axiom,
    uncurr7650761721940715016nt_int = produc4245557441103728435nt_int ).

% uncurry_def
thf(fact_8560_uncurry__def,axiom,
    uncurry_int_int_o = produc4947309494688390418_int_o ).

% uncurry_def
thf(fact_8561_uncurry__def,axiom,
    uncurry_int_int_int = produc8211389475949308722nt_int ).

% uncurry_def
thf(fact_8562_internal__case__prod__def,axiom,
    produc1854806715440696265at_nat = produc27273713700761075at_nat ).

% internal_case_prod_def
thf(fact_8563_internal__case__prod__def,axiom,
    produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).

% internal_case_prod_def
thf(fact_8564_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
thf(fact_8565_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
thf(fact_8566_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
thf(fact_8567_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I2 ) ) ) ) ).

% finite_divisors_int
thf(fact_8568_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_8569_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( ( dvd_dvd_int @ Z @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_8570_dvd__imp__le__int,axiom,
    ! [I2: int,D2: int] :
      ( ( I2 != zero_zero_int )
     => ( ( dvd_dvd_int @ D2 @ I2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ D2 ) @ ( abs_abs_int @ I2 ) ) ) ) ).

% dvd_imp_le_int
thf(fact_8571_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_8572_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_8573_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8574_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8575_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8576_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8577_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_8578_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_8579_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_8580_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_8581_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_8582_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: code_integer,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_8583_bits__induct,axiom,
    ! [P: code_natural > $o,A: code_natural] :
      ( ! [A4: code_natural] :
          ( ( ( divide5121882707175180666atural @ A4 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: code_natural,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B4 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B4 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_8584_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A4: nat] :
          ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: nat,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_8585_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A4: int] :
          ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: int,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_8586_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_8587_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_8588_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q7: nat] : ( product_Pair_nat_nat @ ( suc @ Q7 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8589_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_8590_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_8591_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_8592_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_8593_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_8594_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_8595_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_8596_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8597_take__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8598_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8599_take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8600_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_8601_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_8602_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_8603_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q7: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_8604_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_8605_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_2397367101498566445atural
        @ ( zero_n8403883297036319079atural
          @ ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M )
             != zero_z2226904508553997617atural )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_8606_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_8607_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_8608_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q7: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_8609_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_8610_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q7: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q7 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_8611_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q7: int,R4: int] : ( product_Pair_int_int @ Q7 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_8612_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q7: nat,R4: nat] : ( product_Pair_nat_nat @ Q7 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_8613_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q7: code_integer,R4: code_integer] : ( produc1086072967326762835nteger @ Q7 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_8614_Divides_Oadjust__div__eq,axiom,
    ! [Q6: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q6 @ R3 ) )
      = ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_8615_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8616_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8617_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8618_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8619_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8620_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8621_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ A @ B )
     => ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_8622_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ A @ B )
     => ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_8623_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_8624_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_8625_case__prodI,axiom,
    ! [F: a > produc6653097349344004940it_nat > $o,A: a,B: produc6653097349344004940it_nat] :
      ( ( F @ A @ B )
     => ( produc8860486935419594360_nat_o @ F @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_8626_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_8627_case__prodI2,axiom,
    ! [P3: produc3925858234332021118et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
          ( ( P3
            = ( produc5001842942810119800et_nat @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc1437786849005270451_nat_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8628_case__prodI2,axiom,
    ! [P3: produc2732055786443039994et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
          ( ( P3
            = ( produc2245416461498447860et_nat @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc838355143741117751_nat_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8629_case__prodI2,axiom,
    ! [P3: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc1573362020775583542_int_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8630_case__prodI2,axiom,
    ! [P3: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc2558449545302689196_int_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8631_case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,C: a > produc6653097349344004940it_nat > $o] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc8860486935419594360_nat_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8632_case__prodI2,axiom,
    ! [P3: product_prod_int_int,C: int > int > $o] :
      ( ! [A4: int,B4: int] :
          ( ( P3
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc4947309494688390418_int_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_8633_mem__case__prodI,axiom,
    ! [Z: $o,C: a > produc6653097349344004940it_nat > set_o,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc5350075035600711000_set_o @ C @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8634_mem__case__prodI,axiom,
    ! [Z: nat,C: a > produc6653097349344004940it_nat > set_nat,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc7315013382793309350et_nat @ C @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8635_mem__case__prodI,axiom,
    ! [Z: int,C: a > produc6653097349344004940it_nat > set_int,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc3137162363284112642et_int @ C @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8636_mem__case__prodI,axiom,
    ! [Z: set_nat,C: a > produc6653097349344004940it_nat > set_set_nat,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_set_nat @ Z @ ( C @ A @ B ) )
     => ( member_set_nat @ Z @ ( produc8632796428668088668et_nat @ C @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8637_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: a > produc6653097349344004940it_nat > set_Pr1261947904930325089at_nat,A: a,B: produc6653097349344004940it_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8253449849199807809at_nat @ C @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8638_mem__case__prodI,axiom,
    ! [Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8639_mem__case__prodI,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8640_mem__case__prodI,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8641_mem__case__prodI,axiom,
    ! [Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_set_nat @ Z @ ( C @ A @ B ) )
     => ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8642_mem__case__prodI,axiom,
    ! [Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_8643_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z: $o,C: a > produc6653097349344004940it_nat > set_o] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_o @ Z @ ( produc5350075035600711000_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8644_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z: nat,C: a > produc6653097349344004940it_nat > set_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc7315013382793309350et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8645_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z: int,C: a > produc6653097349344004940it_nat > set_int] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_int @ Z @ ( produc3137162363284112642et_int @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8646_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z: set_nat,C: a > produc6653097349344004940it_nat > set_set_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( member_set_nat @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_set_nat @ Z @ ( produc8632796428668088668et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8647_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z: product_prod_nat_nat,C: a > produc6653097349344004940it_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A4: a,B4: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A4 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8253449849199807809at_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8648_mem__case__prodI2,axiom,
    ! [P3: produc7773217078559923341nt_int,Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8649_mem__case__prodI2,axiom,
    ! [P3: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8650_mem__case__prodI2,axiom,
    ! [P3: produc7773217078559923341nt_int,Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8651_mem__case__prodI2,axiom,
    ! [P3: produc7773217078559923341nt_int,Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P3
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_set_nat @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8652_mem__case__prodI2,axiom,
    ! [P3: produc3925858234332021118et_nat,Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o] :
      ( ! [A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
          ( ( P3
            = ( produc5001842942810119800et_nat @ A4 @ B4 ) )
         => ( member_o @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_8653_case__prodI2_H,axiom,
    ! [P3: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ( product_Pair_nat_nat @ A4 @ B4 )
            = P3 )
         => ( C @ A4 @ B4 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P3 @ X ) ) ).

% case_prodI2'
thf(fact_8654_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_8655_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_8656_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_8657_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_8658_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_8659_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_8660_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( case_o356765784539232260er_num @ zero_z3403309356797280102nteger @ numera6620942414471956472nteger @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_8661_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( case_o5621594795226839503al_num @ zero_z2226904508553997617atural @ numera5444537566228673987atural @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_8662_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( case_option_nat_num @ zero_zero_nat @ numeral_numeral_nat @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_8663_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_8664_mem__case__prodE,axiom,
    ! [Z: $o,C: a > produc6653097349344004940it_nat > set_o,P3: produc3260487557148687353it_nat] :
      ( ( member_o @ Z @ ( produc5350075035600711000_set_o @ C @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8665_mem__case__prodE,axiom,
    ! [Z: nat,C: a > produc6653097349344004940it_nat > set_nat,P3: produc3260487557148687353it_nat] :
      ( ( member_nat @ Z @ ( produc7315013382793309350et_nat @ C @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8666_mem__case__prodE,axiom,
    ! [Z: int,C: a > produc6653097349344004940it_nat > set_int,P3: produc3260487557148687353it_nat] :
      ( ( member_int @ Z @ ( produc3137162363284112642et_int @ C @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8667_mem__case__prodE,axiom,
    ! [Z: set_nat,C: a > produc6653097349344004940it_nat > set_set_nat,P3: produc3260487557148687353it_nat] :
      ( ( member_set_nat @ Z @ ( produc8632796428668088668et_nat @ C @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_set_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8668_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: a > produc6653097349344004940it_nat > set_Pr1261947904930325089at_nat,P3: produc3260487557148687353it_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc8253449849199807809at_nat @ C @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8669_mem__case__prodE,axiom,
    ! [Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o,P3: produc7773217078559923341nt_int] :
      ( ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ P3 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8670_mem__case__prodE,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P3: produc7773217078559923341nt_int] :
      ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P3 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8671_mem__case__prodE,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,P3: produc7773217078559923341nt_int] :
      ( ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P3 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8672_mem__case__prodE,axiom,
    ! [Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat,P3: produc7773217078559923341nt_int] :
      ( ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ P3 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( member_set_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8673_mem__case__prodE,axiom,
    ! [Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o,P3: produc3925858234332021118et_nat] :
      ( ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ P3 ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P3
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_8674_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8675_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8676_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8677_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8678_case__prodD,axiom,
    ! [F: a > produc6653097349344004940it_nat > $o,A: a,B: produc6653097349344004940it_nat] :
      ( ( produc8860486935419594360_nat_o @ F @ ( produc9178034014595674355it_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8679_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_8680_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
      ( ( produc1437786849005270451_nat_o @ C @ P3 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P3
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8681_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,P3: produc2732055786443039994et_nat] :
      ( ( produc838355143741117751_nat_o @ C @ P3 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
            ( ( P3
              = ( produc2245416461498447860et_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8682_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P3: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P3 )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8683_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P3: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P3 )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( P3
              = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8684_case__prodE,axiom,
    ! [C: a > produc6653097349344004940it_nat > $o,P3: produc3260487557148687353it_nat] :
      ( ( produc8860486935419594360_nat_o @ C @ P3 )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8685_case__prodE,axiom,
    ! [C: int > int > $o,P3: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P3 )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_8686_case__prodD_H,axiom,
    ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_8687_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P3 @ Z )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 @ Z ) ) ) ).

% case_prodE'
thf(fact_8688_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q7: int,R4: int] : ( plus_plus_int @ Q7 @ ( zero_n2684676970156552555ol_int @ ( R4 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_8689_Collect__case__prod__mono,axiom,
    ! [A3: int > int > $o,B3: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A3 @ B3 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A3 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B3 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_8690_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_8691_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8692_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8693_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_a,G: a > heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_a @ ( heap_Time_bind_a_a @ F @ G ) @ H )
      = ( case_o6061097939050036047it_nat @ none_P2651198173097904984it_nat
        @ ( produc7850007024774191849it_nat
          @ ^ [X4: a] :
              ( produc5645566021302314940it_nat
              @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711335rame_a @ N @ ( heap_Time_execute_a @ ( G @ X4 ) @ H7 ) ) ) )
        @ ( heap_Time_execute_a @ F @ H ) ) ) ).

% execute_bind_case
thf(fact_8694_map__to__set__def,axiom,
    ( map_to_set_int_int
    = ( ^ [M3: int > option_int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [K3: int,V2: int] :
                ( ( M3 @ K3 )
                = ( some_int @ V2 ) ) ) ) ) ) ).

% map_to_set_def
thf(fact_8695_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q6: num] :
      ( ( ( bit_take_bit_num @ M @ N2 )
        = ( some_num @ Q6 ) )
     => ( ( bit_se1745604003318907178nteger @ M @ ( numera6620942414471956472nteger @ N2 ) )
        = ( numera6620942414471956472nteger @ Q6 ) ) ) ).

% take_bit_num_eq_Some_imp
thf(fact_8696_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q6: num] :
      ( ( ( bit_take_bit_num @ M @ N2 )
        = ( some_num @ Q6 ) )
     => ( ( bit_se569199155075624693atural @ M @ ( numera5444537566228673987atural @ N2 ) )
        = ( numera5444537566228673987atural @ Q6 ) ) ) ).

% take_bit_num_eq_Some_imp
thf(fact_8697_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q6: num] :
      ( ( ( bit_take_bit_num @ M @ N2 )
        = ( some_num @ Q6 ) )
     => ( ( bit_se2925701944663578781it_nat @ M @ ( numeral_numeral_nat @ N2 ) )
        = ( numeral_numeral_nat @ Q6 ) ) ) ).

% take_bit_num_eq_Some_imp
thf(fact_8698_take__bit__num__eq__Some__imp,axiom,
    ! [M: nat,N2: num,Q6: num] :
      ( ( ( bit_take_bit_num @ M @ N2 )
        = ( some_num @ Q6 ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( numeral_numeral_int @ N2 ) )
        = ( numeral_numeral_int @ Q6 ) ) ) ).

% take_bit_num_eq_Some_imp
thf(fact_8699_Heap__Time__Monad_Obind__def,axiom,
    ( heap_Time_bind_a_a
    = ( ^ [F3: heap_Time_Heap_a,G3: a > heap_Time_Heap_a] :
          ( heap_Time_Heap_a2
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o6061097939050036047it_nat @ none_P2651198173097904984it_nat
              @ ( produc7850007024774191849it_nat
                @ ^ [R4: a] :
                    ( produc5645566021302314940it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711335rame_a @ N @ ( heap_Time_execute_a @ ( G3 @ R4 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_a @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_8700_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T8685611227969916822t_unit
    = ( ^ [F3: heap_Time_Heap_a,G3: a > heap_T5738788834812785303t_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o7727800614283616222it_nat @ none_P9117596204409417319it_nat
              @ ( produc5648861294512138360it_nat
                @ ^ [R4: a] :
                    ( produc7488178964372371019it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T3616969660504097270t_unit @ N @ ( heap_T875086893843062177t_unit @ ( G3 @ R4 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_a @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_8701_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A6: set_nat,B6: set_nat] :
            ( ( ord_less_set_nat @ A6 @ B6 )
            & ( finite_finite_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8702_finite__psubset__def,axiom,
    ( finite_psubset_int
    = ( collec957716948307931664et_int
      @ ( produc4109468873575309990_int_o
        @ ^ [A6: set_int,B6: set_int] :
            ( ( ord_less_set_int @ A6 @ B6 )
            & ( finite_finite_int @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8703_finite__psubset__def,axiom,
    ( finite2416775604798480986nteger
    = ( collec2770208431294612722nteger
      @ ( produc1658495936642081476eger_o
        @ ^ [A6: set_Code_integer,B6: set_Code_integer] :
            ( ( ord_le1307284697595431911nteger @ A6 @ B6 )
            & ( finite6017078050557962740nteger @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8704_finite__psubset__def,axiom,
    ( finite469560695537375940at_nat
    = ( collec6321179662152712658at_nat
      @ ( produc410239310623530412_nat_o
        @ ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( ord_le7866589430770878221at_nat @ A6 @ B6 )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8705_rel__of__def,axiom,
    ( rel_of4838799251197538391et_nat
    = ( ^ [M3: ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat,P4: produc3925858234332021118et_nat > $o] :
          ( collec1402215087704437587et_nat
          @ ( produc1437786849005270451_nat_o
            @ ^ [K3: produc3658429121746597890et_nat > $o,V2: produc3658429121746597890et_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P624177172695371229et_nat @ V2 ) )
                & ( P4 @ ( produc5001842942810119800et_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8706_rel__of__def,axiom,
    ( rel_of7774016450764239315et_nat
    = ( ^ [M3: ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat,P4: produc2732055786443039994et_nat > $o] :
          ( collec5543584681430388431et_nat
          @ ( produc838355143741117751_nat_o
            @ ^ [K3: produc3658429121746597890et_nat > $o,V2: produc3925858234332021118et_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P750831030444334937et_nat @ V2 ) )
                & ( P4 @ ( produc2245416461498447860et_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8707_rel__of__def,axiom,
    ( rel_of8306664904814525588nt_int
    = ( ^ [M3: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int,P4: produc2285326912895808259nt_int > $o] :
          ( collec1790188477890212312nt_int
          @ ( produc1573362020775583542_int_o
            @ ^ [K3: produc8551481072490612790e_term > option6357759511663192854e_term,V2: product_prod_int_int] :
                ( ( ( M3 @ K3 )
                  = ( some_P4184893108420464158nt_int @ V2 ) )
                & ( P4 @ ( produc5700946648718959541nt_int @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8708_rel__of__def,axiom,
    ( rel_of5543720577181062686nt_int
    = ( ^ [M3: ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int,P4: produc7773217078559923341nt_int > $o] :
          ( collec506566255779805410nt_int
          @ ( produc2558449545302689196_int_o
            @ ^ [K3: int > option6357759511663192854e_term,V2: product_prod_int_int] :
                ( ( ( M3 @ K3 )
                  = ( some_P4184893108420464158nt_int @ V2 ) )
                & ( P4 @ ( produc4305682042979456191nt_int @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8709_rel__of__def,axiom,
    ( rel_of2761222537318653906it_nat
    = ( ^ [M3: a > option233860712434008220it_nat,P4: produc3260487557148687353it_nat > $o] :
          ( collec4148678073425210574it_nat
          @ ( produc8860486935419594360_nat_o
            @ ^ [K3: a,V2: produc6653097349344004940it_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P1054239925786823975it_nat @ V2 ) )
                & ( P4 @ ( produc9178034014595674355it_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8710_rel__of__def,axiom,
    ( rel_of_int_int
    = ( ^ [M3: int > option_int,P4: product_prod_int_int > $o] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [K3: int,V2: int] :
                ( ( ( M3 @ K3 )
                  = ( some_int @ V2 ) )
                & ( P4 @ ( product_Pair_int_int @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_8711_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_8712_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_8713_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_8714_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_8715_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_8716_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F3: nat > nat,R6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_8717_mlex__eq,axiom,
    ( mlex_prod_int
    = ( ^ [F3: int > nat,R6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [X4: int,Y4: int] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_8718_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q7: int,R4: int] : ( product_Pair_int_int @ Q7 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_8719_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q7: nat,R4: nat] : ( product_Pair_nat_nat @ Q7 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_8720_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q7: code_integer,R4: code_integer] : ( produc1086072967326762835nteger @ Q7 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R4 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_8721_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z6: int] :
                ( ( ord_less_eq_int @ D5 @ Z6 )
                & ( ord_less_int @ Z8 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8722_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z6: int] :
                ( ( ord_less_eq_int @ D5 @ Z8 )
                & ( ord_less_int @ Z8 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8723_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8724_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8725_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8726_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A5: int,B5: int] :
            ( P
            & ( Q @ A5 @ B5 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_8727_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_8728_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(73)
thf(fact_8729_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(72)
thf(fact_8730_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_8731_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_8732_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_8733_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_8734_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_8735_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_8736_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8737_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8738_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8739_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8740_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8741_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8742_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu2: int,Uv: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_8743_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 ) ) )
         => ( ! [M5: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
           => ( ! [M5: num,N5: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) )
             => ( ! [M5: num,N5: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) )
               => ( ! [M5: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
                 => ( ! [M5: num,N5: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) )
                   => ~ ! [M5: num,N5: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_8744_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_8745_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_8746_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ one_one_Code_integer ) ) ).

% numeral_code(3)
thf(fact_8747_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_8748_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) @ one_one_Code_natural ) ) ).

% numeral_code(3)
thf(fact_8749_power__numeral__odd,axiom,
    ! [Z: code_integer,W2: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8750_power__numeral__odd,axiom,
    ! [Z: assn,W2: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_assn @ ( times_times_assn @ Z @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8751_power__numeral__odd,axiom,
    ! [Z: rat,W2: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8752_power__numeral__odd,axiom,
    ! [Z: nat,W2: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8753_power__numeral__odd,axiom,
    ! [Z: int,W2: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8754_power3__eq__cube,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_8755_power3__eq__cube,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_assn @ ( times_times_assn @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_8756_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_8757_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_8758_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_8759_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_8760_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_8761_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_8762_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_8763_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_8764_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_8765_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_bit1
thf(fact_8766_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_numeral_bit1
thf(fact_8767_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_8768_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_8769_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8770_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8771_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_8772_lessThan__iff,axiom,
    ! [I2: $o,K: $o] :
      ( ( member_o @ I2 @ ( set_ord_lessThan_o @ K ) )
      = ( ord_less_o @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8773_lessThan__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8774_lessThan__iff,axiom,
    ! [I2: assn,K: assn] :
      ( ( member_assn @ I2 @ ( set_or7637083652282234053n_assn @ K ) )
      = ( ord_less_assn @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8775_lessThan__iff,axiom,
    ! [I2: rat,K: rat] :
      ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8776_lessThan__iff,axiom,
    ! [I2: num,K: num] :
      ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8777_lessThan__iff,axiom,
    ! [I2: nat,K: nat] :
      ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8778_lessThan__iff,axiom,
    ! [I2: int,K: int] :
      ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_8779_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8780_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8781_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8782_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8783_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8784_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_8785_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_8786_prod_OlessThan__Suc,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8787_prod_OlessThan__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8788_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8789_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8790_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8791_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_8792_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_8793_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_8794_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_8795_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8796_lessThan__def,axiom,
    ( set_or7637083652282234053n_assn
    = ( ^ [U2: assn] :
          ( collect_assn
          @ ^ [X4: assn] : ( ord_less_assn @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8797_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_rat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8798_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_num @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8799_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8800_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8801_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8802_Iio__eq__empty__iff,axiom,
    ! [N2: $o] :
      ( ( ( set_ord_lessThan_o @ N2 )
        = bot_bot_set_o )
      = ( N2 = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_8803_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_8804_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S5: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8805_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_8806_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8807_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8808_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8809_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_8810_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8811_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8812_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8813_ivl__disj__un__one_I2_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or4029947393144176647an_rat @ L @ U ) )
        = ( set_ord_lessThan_rat @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_8814_ivl__disj__un__one_I2_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_lessThan_num @ L ) @ ( set_or1222409239386451017an_num @ L @ U ) )
        = ( set_ord_lessThan_num @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_8815_ivl__disj__un__one_I2_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
        = ( set_ord_lessThan_nat @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_8816_ivl__disj__un__one_I2_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or4662586982721622107an_int @ L @ U ) )
        = ( set_ord_lessThan_int @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_8817_ivl__disj__un__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or8404916559141939852nteger @ L @ U ) )
        = ( set_or5754767410780653050nteger @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_8818_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(4)
thf(fact_8819_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(4)
thf(fact_8820_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(4)
thf(fact_8821_ivl__disj__int__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( inf_in1364745209274528805nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or189985376899183464nteger @ L @ U ) )
      = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_one(4)
thf(fact_8822_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8823_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8824_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8825_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8826_ivl__disj__int__one_I2_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or7139685690850216873Than_o @ L @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(2)
thf(fact_8827_ivl__disj__int__one_I2_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(2)
thf(fact_8828_ivl__disj__int__one_I2_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or4662586982721622107an_int @ L @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(2)
thf(fact_8829_ivl__disj__int__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( inf_in1364745209274528805nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or8404916559141939852nteger @ L @ U ) )
      = bot_bo3990330152332043303nteger ) ).

% ivl_disj_int_one(2)
thf(fact_8830_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N2: int] :
      ( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8831_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8832_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8833_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8834_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8835_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8836_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8837_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8838_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8839_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8840_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( G @ zero_zero_nat )
        @ ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8841_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8842_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8843_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8844_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8845_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_8846_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8847_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_8848_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_8849_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_8850_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8851_sum_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [M3: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% sum.nat_group
thf(fact_8852_prod_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [M3: nat] : ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_8853_prod_Onat__group,axiom,
    ! [G: nat > int,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [M3: nat] : ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_8854_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8855_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8856_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( groups705719431365010083at_int @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8857_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8858_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ K ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ K ) @ ( semiri3624122377584611663nteger @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8859_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8860_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2447717529341329178atural @ ( numeral_numeral_nat @ K ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ K ) @ ( semiri2447717529341329178atural @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8861_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8862_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
        = ( set_ord_atMost_rat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_8863_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_lessThan_num @ L ) @ ( set_or7049704709247886629st_num @ L @ U ) )
        = ( set_ord_atMost_num @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_8864_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
        = ( set_ord_atMost_int @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_8865_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
        = ( set_ord_atMost_nat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_8866_ivl__disj__un__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or189985376899183464nteger @ L @ U ) )
        = ( set_or9101266186257409494nteger @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_8867_one__diff__power__eq,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8868_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8869_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8870_power__diff__1__eq,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8871_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8872_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8873_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_8874_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_8875_sum_OatMost__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8876_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8877_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8878_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8879_prod_OatMost__shift,axiom,
    ! [G: nat > code_integer,N2: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_3573771949741848930nteger @ ( G @ zero_zero_nat )
        @ ( groups3455450783089532116nteger
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8880_prod_OatMost__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8881_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8882_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8883_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8884_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_8885_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_8886_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8887_diff__power__eq__sum,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) @ ( power_8256067586552552935nteger @ Y @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [P7: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ P7 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N2 @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8888_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P7: nat] : ( times_times_rat @ ( power_power_rat @ X @ P7 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8889_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P7: nat] : ( times_times_int @ ( power_power_int @ X @ P7 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P7 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8890_power__diff__sumr2,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) ) @ ( power_8256067586552552935nteger @ X @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8891_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8892_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) ) @ ( power_power_int @ X @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8893_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_8894_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_8895_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_8896_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_8897_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
        @ ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8898_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8899_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8900_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_8901_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_8902_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_8903_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_8904_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_8905_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_8906_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_8907_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_8908_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_8909_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_8910_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_8911_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_8912_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_8913_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_8914_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q7: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q7 ) @ R4 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_8915_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_8916_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_8917_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_8918_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_8919_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_8920_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_8921_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_8922_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_8923_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_8924_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_8925_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_8926_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_8927_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D: code_integer,I3: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_8928_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D: code_integer,I3: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_8929_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X4: code_integer] : ( plus_p5714425477246183910nteger @ X4 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_8930_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_8931_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_8932_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_8933_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_8934_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_8935_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_8936_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_8937_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_8938_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_8939_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S5: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_8940_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_8941_horner__sum__eq__sum,axiom,
    ( groups8594936613892548326nteger
    = ( ^ [F3: int > code_integer,A5: code_integer,Xs3: list_int] :
          ( groups7501900531339628137nteger
          @ ^ [N: nat] : ( times_3573771949741848930nteger @ ( F3 @ ( nth_int @ Xs3 @ N ) ) @ ( power_8256067586552552935nteger @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8942_horner__sum__eq__sum,axiom,
    ( groups8223283053655703690nteger
    = ( ^ [F3: nat > code_integer,A5: code_integer,Xs3: list_nat] :
          ( groups7501900531339628137nteger
          @ ^ [N: nat] : ( times_3573771949741848930nteger @ ( F3 @ ( nth_nat @ Xs3 @ N ) ) @ ( power_8256067586552552935nteger @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8943_horner__sum__eq__sum,axiom,
    ( groups3417619833198082522nteger
    = ( ^ [F3: $o > code_integer,A5: code_integer,Xs3: list_o] :
          ( groups7501900531339628137nteger
          @ ^ [N: nat] : ( times_3573771949741848930nteger @ ( F3 @ ( nth_o @ Xs3 @ N ) ) @ ( power_8256067586552552935nteger @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8944_horner__sum__eq__sum,axiom,
    ( groups7852591826665563233nt_rat
    = ( ^ [F3: int > rat,A5: rat,Xs3: list_int] :
          ( groups2906978787729119204at_rat
          @ ^ [N: nat] : ( times_times_rat @ ( F3 @ ( nth_int @ Xs3 @ N ) ) @ ( power_power_rat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8945_horner__sum__eq__sum,axiom,
    ( groups6853238114764508677at_rat
    = ( ^ [F3: nat > rat,A5: rat,Xs3: list_nat] :
          ( groups2906978787729119204at_rat
          @ ^ [N: nat] : ( times_times_rat @ ( F3 @ ( nth_nat @ Xs3 @ N ) ) @ ( power_power_rat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8946_horner__sum__eq__sum,axiom,
    ( groups8483887719401441109_o_rat
    = ( ^ [F3: $o > rat,A5: rat,Xs3: list_o] :
          ( groups2906978787729119204at_rat
          @ ^ [N: nat] : ( times_times_rat @ ( F3 @ ( nth_o @ Xs3 @ N ) ) @ ( power_power_rat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8947_horner__sum__eq__sum,axiom,
    ( groups8485231416243008693nt_int
    = ( ^ [F3: int > int,A5: int,Xs3: list_int] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F3 @ ( nth_int @ Xs3 @ N ) ) @ ( power_power_int @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8948_horner__sum__eq__sum,axiom,
    ( groups7485877704341954137at_int
    = ( ^ [F3: nat > int,A5: int,Xs3: list_nat] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F3 @ ( nth_nat @ Xs3 @ N ) ) @ ( power_power_int @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8949_horner__sum__eq__sum,axiom,
    ( groups8487721886752058969nt_nat
    = ( ^ [F3: int > nat,A5: nat,Xs3: list_int] :
          ( groups3542108847815614940at_nat
          @ ^ [N: nat] : ( times_times_nat @ ( F3 @ ( nth_int @ Xs3 @ N ) ) @ ( power_power_nat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8950_horner__sum__eq__sum,axiom,
    ( groups7488368174851004413at_nat
    = ( ^ [F3: nat > nat,A5: nat,Xs3: list_nat] :
          ( groups3542108847815614940at_nat
          @ ^ [N: nat] : ( times_times_nat @ ( F3 @ ( nth_nat @ Xs3 @ N ) ) @ ( power_power_nat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8951_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_8952_integer__of__num_I3_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_8953_take__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M3: num] :
          ( produc478579273971653890on_num
          @ ^ [A5: nat,X4: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P7: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q7: num] : ( some_num @ ( bit0 @ Q7 ) )
                      @ ( bit_take_bit_num @ O @ P7 ) )
                  @ ^ [P7: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P7 ) ) )
                  @ X4 )
              @ A5 )
          @ ( product_Pair_nat_num @ N @ M3 ) ) ) ) ).

% take_bit_num_code
thf(fact_8954_even__sum__iff,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ( finite_finite_o @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups8507830703676809646_o_nat @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A3 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8955_even__sum__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_int
            @ ( collect_int
              @ ^ [A5: int] :
                  ( ( member_int @ A5 @ A3 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8956_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups7237345082560585321er_nat @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A3 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8957_even__sum__iff,axiom,
    ! [A3: set_o,F: $o > int] :
      ( ( finite_finite_o @ A3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups8505340233167759370_o_int @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8958_even__sum__iff,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [A5: nat] :
                  ( ( member_nat @ A5 @ A3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8959_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups7234854612051535045er_int @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8960_even__sum__iff,axiom,
    ! [A3: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ A3 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups4406642042086082107nteger @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A3 )
                  & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8961_even__sum__iff,axiom,
    ! [A3: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [A5: nat] :
                  ( ( member_nat @ A5 @ A3 )
                  & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8962_even__sum__iff,axiom,
    ! [A3: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ A3 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7873554091576472773nteger @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_int
            @ ( collect_int
              @ ^ [A5: int] :
                  ( ( member_int @ A5 @ A3 )
                  & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8963_even__sum__iff,axiom,
    ! [A3: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups879477027807139574nteger @ F @ A3 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A3 )
                  & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_8964_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_8965_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_8966_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_8967_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_8968_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8969_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8970_card_Oempty,axiom,
    ( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8971_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8972_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8973_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8974_prod__constant,axiom,
    ! [Y: nat,A3: set_set_nat] :
      ( ( groups4248547760180025341at_nat
        @ ^ [X4: set_nat] : Y
        @ A3 )
      = ( power_power_nat @ Y @ ( finite_card_set_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8975_prod__constant,axiom,
    ! [Y: nat,A3: set_list_nat] :
      ( ( groups2907647131375434839at_nat
        @ ^ [X4: list_nat] : Y
        @ A3 )
      = ( power_power_nat @ Y @ ( finite_card_list_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8976_prod__constant,axiom,
    ! [Y: int,A3: set_set_nat] :
      ( ( groups4246057289670975065at_int
        @ ^ [X4: set_nat] : Y
        @ A3 )
      = ( power_power_int @ Y @ ( finite_card_set_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8977_prod__constant,axiom,
    ! [Y: int,A3: set_list_nat] :
      ( ( groups2905156660866384563at_int
        @ ^ [X4: list_nat] : Y
        @ A3 )
      = ( power_power_int @ Y @ ( finite_card_list_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8978_prod__constant,axiom,
    ! [Y: nat,A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( power_power_nat @ Y @ ( finite_card_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8979_prod__constant,axiom,
    ! [Y: int,A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( power_power_int @ Y @ ( finite_card_nat @ A3 ) ) ) ).

% prod_constant
thf(fact_8980_prod__constant,axiom,
    ! [Y: int,A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : Y
        @ A3 )
      = ( power_power_int @ Y @ ( finite_card_int @ A3 ) ) ) ).

% prod_constant
thf(fact_8981_case__nat__numeral,axiom,
    ! [A: option_num,F: nat > option_num,V: num] :
      ( ( case_nat_option_num @ A @ F @ ( numeral_numeral_nat @ V ) )
      = ( F @ ( pred_numeral @ V ) ) ) ).

% case_nat_numeral
thf(fact_8982_case__nat__numeral,axiom,
    ! [A: $o,F: nat > $o,V: num] :
      ( ( case_nat_o @ A @ F @ ( numeral_numeral_nat @ V ) )
      = ( F @ ( pred_numeral @ V ) ) ) ).

% case_nat_numeral
thf(fact_8983_case__nat__numeral,axiom,
    ! [A: nat,F: nat > nat,V: num] :
      ( ( case_nat_nat @ A @ F @ ( numeral_numeral_nat @ V ) )
      = ( F @ ( pred_numeral @ V ) ) ) ).

% case_nat_numeral
thf(fact_8984_card__0__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ( finite_card_set_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_8985_card__0__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( ( finite_card_list_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_8986_card__0__eq,axiom,
    ! [A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( ( finite4902975817058060853nteger @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bo3990330152332043303nteger ) ) ) ).

% card_0_eq
thf(fact_8987_card__0__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( finite711546835091564841at_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
thf(fact_8988_card__0__eq,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( finite_card_o @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_8989_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( finite_card_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_8990_card__0__eq,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( finite_card_int @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_8991_sum__constant,axiom,
    ! [Y: rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8992_sum__constant,axiom,
    ! [Y: int,A3: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8993_sum__constant,axiom,
    ! [Y: code_integer,A3: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8994_sum__constant,axiom,
    ! [Y: nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : Y
        @ A3 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8995_sum__constant,axiom,
    ! [Y: int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : Y
        @ A3 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_int @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8996_sum__constant,axiom,
    ! [Y: rat,A3: set_set_nat] :
      ( ( groups7659867448343625626at_rat
        @ ^ [X4: set_nat] : Y
        @ A3 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_set_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8997_sum__constant,axiom,
    ! [Y: rat,A3: set_list_nat] :
      ( ( groups3760926236672600436at_rat
        @ ^ [X4: list_nat] : Y
        @ A3 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_list_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8998_sum__constant,axiom,
    ! [Y: int,A3: set_set_nat] :
      ( ( groups8292507037921071086at_int
        @ ^ [X4: set_nat] : Y
        @ A3 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_set_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_8999_sum__constant,axiom,
    ! [Y: int,A3: set_list_nat] :
      ( ( groups4393565826250045896at_int
        @ ^ [X4: list_nat] : Y
        @ A3 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9000_sum__constant,axiom,
    ! [Y: nat,A3: set_set_nat] :
      ( ( groups8294997508430121362at_nat
        @ ^ [X4: set_nat] : Y
        @ A3 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_set_nat @ A3 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9001_case__nat__add__eq__if,axiom,
    ! [A: option_num,F: nat > option_num,V: num,N2: nat] :
      ( ( case_nat_option_num @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_9002_case__nat__add__eq__if,axiom,
    ! [A: $o,F: nat > $o,V: num,N2: nat] :
      ( ( case_nat_o @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_9003_case__nat__add__eq__if,axiom,
    ! [A: nat,F: nat > nat,V: num,N2: nat] :
      ( ( case_nat_nat @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_9004_sum__of__bool__eq,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups7873554091576472773nteger
            @ ^ [X4: int] : ( zero_n356916108424825756nteger @ ( P @ X4 ) )
            @ A3 )
          = ( semiri4939895301339042750nteger @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9005_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups879477027807139574nteger
            @ ^ [X4: code_integer] : ( zero_n356916108424825756nteger @ ( P @ X4 ) )
            @ A3 )
          = ( semiri4939895301339042750nteger @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9006_sum__of__bool__eq,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups7501900531339628137nteger
            @ ^ [X4: nat] : ( zero_n356916108424825756nteger @ ( P @ X4 ) )
            @ A3 )
          = ( semiri4939895301339042750nteger @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9007_sum__of__bool__eq,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] : ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1316708129612266289at_nat @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9008_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X4: code_integer] : ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1316708129612266289at_nat @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9009_sum__of__bool__eq,axiom,
    ! [A3: set_Code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ A3 )
       => ( ( groups7234854612051535045er_int
            @ ^ [X4: code_integer] : ( zero_n2684676970156552555ol_int @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1314217659103216013at_int @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A3 @ ( collect_Code_integer @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9010_sum__of__bool__eq,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X4: nat] : ( zero_n2684676970156552555ol_int @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1314217659103216013at_int @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9011_sum__of__bool__eq,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] : ( zero_n2687167440665602831ol_nat @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1316708129612266289at_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9012_sum__of__bool__eq,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups4538972089207619220nt_int
            @ ^ [X4: int] : ( zero_n2684676970156552555ol_int @ ( P @ X4 ) )
            @ A3 )
          = ( semiri1314217659103216013at_int @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9013_sum__of__bool__eq,axiom,
    ! [A3: set_set_nat,P: set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ A3 )
       => ( ( groups9190459664516455967nteger
            @ ^ [X4: set_nat] : ( zero_n356916108424825756nteger @ ( P @ X4 ) )
            @ A3 )
          = ( semiri4939895301339042750nteger @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9014_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_9015_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9016_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa @ X ) ) ).

% less_integer.abs_eq
thf(fact_9017_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9018_nat_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F22: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9019_nat_Ocase__distrib,axiom,
    ! [H: option_num > $o,F1: option_num,F22: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9020_nat_Ocase__distrib,axiom,
    ! [H: option_num > nat,F1: option_num,F22: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9021_nat_Ocase__distrib,axiom,
    ! [H: $o > option_num,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9022_nat_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9023_nat_Ocase__distrib,axiom,
    ! [H: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9024_nat_Ocase__distrib,axiom,
    ! [H: nat > option_num,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9025_nat_Ocase__distrib,axiom,
    ! [H: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9026_nat_Ocase__distrib,axiom,
    ! [H: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_9027_num_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F22: num > option_num,F32: num > option_num,Num: num] :
      ( ( H @ ( case_num_option_num @ F1 @ F22 @ F32 @ Num ) )
      = ( case_num_option_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
        @ ^ [X4: num] : ( H @ ( F32 @ X4 ) )
        @ Num ) ) ).

% num.case_distrib
thf(fact_9028_n__subsets,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1149291290879098388et_nat
          @ ( collect_set_set_nat
            @ ^ [B6: set_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ B6 @ A3 )
                & ( ( finite_card_set_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_set_nat @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9029_n__subsets,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite2364142230527598318st_nat
          @ ( collect_set_list_nat
            @ ^ [B6: set_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ B6 @ A3 )
                & ( ( finite_card_list_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_list_nat @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9030_n__subsets,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [B6: set_nat] :
                ( ( ord_less_eq_set_nat @ B6 @ A3 )
                & ( ( finite_card_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_nat @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9031_n__subsets,axiom,
    ! [A3: set_Code_integer,K: nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite1340570857867686763nteger
          @ ( collec574505750873337192nteger
            @ ^ [B6: set_Code_integer] :
                ( ( ord_le7084787975880047091nteger @ B6 @ A3 )
                & ( ( finite4902975817058060853nteger @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite4902975817058060853nteger @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9032_n__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite4356350796350151305at_nat
          @ ( collec5514110066124741708at_nat
            @ ^ [B6: set_Pr1261947904930325089at_nat] :
                ( ( ord_le3146513528884898305at_nat @ B6 @ A3 )
                & ( ( finite711546835091564841at_nat @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite711546835091564841at_nat @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9033_n__subsets,axiom,
    ! [A3: set_int,K: nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_card_set_int
          @ ( collect_set_int
            @ ^ [B6: set_int] :
                ( ( ord_less_eq_set_int @ B6 @ A3 )
                & ( ( finite_card_int @ B6 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_int @ A3 ) @ K ) ) ) ).

% n_subsets
thf(fact_9034_old_Onat_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: nat > option_num] :
      ( ( case_nat_option_num @ F1 @ F22 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_9035_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F22: nat > $o] :
      ( ( case_nat_o @ F1 @ F22 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_9036_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F22: nat > nat] :
      ( ( case_nat_nat @ F1 @ F22 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_9037_old_Onat_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: nat > option_num,X2: nat] :
      ( ( case_nat_option_num @ F1 @ F22 @ ( suc @ X2 ) )
      = ( F22 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_9038_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F22: nat > $o,X2: nat] :
      ( ( case_nat_o @ F1 @ F22 @ ( suc @ X2 ) )
      = ( F22 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_9039_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F22: nat > nat,X2: nat] :
      ( ( case_nat_nat @ F1 @ F22 @ ( suc @ X2 ) )
      = ( F22 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_9040_infinite__arbitrarily__large,axiom,
    ! [A3: set_set_nat,N2: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A3 )
     => ? [B10: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B10 )
          & ( ( finite_card_set_nat @ B10 )
            = N2 )
          & ( ord_le6893508408891458716et_nat @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9041_infinite__arbitrarily__large,axiom,
    ! [A3: set_list_nat,N2: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A3 )
     => ? [B10: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B10 )
          & ( ( finite_card_list_nat @ B10 )
            = N2 )
          & ( ord_le6045566169113846134st_nat @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9042_infinite__arbitrarily__large,axiom,
    ! [A3: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [B10: set_nat] :
          ( ( finite_finite_nat @ B10 )
          & ( ( finite_card_nat @ B10 )
            = N2 )
          & ( ord_less_eq_set_nat @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9043_infinite__arbitrarily__large,axiom,
    ! [A3: set_Code_integer,N2: nat] :
      ( ~ ( finite6017078050557962740nteger @ A3 )
     => ? [B10: set_Code_integer] :
          ( ( finite6017078050557962740nteger @ B10 )
          & ( ( finite4902975817058060853nteger @ B10 )
            = N2 )
          & ( ord_le7084787975880047091nteger @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9044_infinite__arbitrarily__large,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ? [B10: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ B10 )
          & ( ( finite711546835091564841at_nat @ B10 )
            = N2 )
          & ( ord_le3146513528884898305at_nat @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9045_infinite__arbitrarily__large,axiom,
    ! [A3: set_int,N2: nat] :
      ( ~ ( finite_finite_int @ A3 )
     => ? [B10: set_int] :
          ( ( finite_finite_int @ B10 )
          & ( ( finite_card_int @ B10 )
            = N2 )
          & ( ord_less_eq_set_int @ B10 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9046_card__subset__eq,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
       => ( ( ( finite_card_set_nat @ A3 )
            = ( finite_card_set_nat @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9047_card__subset__eq,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B3 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
       => ( ( ( finite_card_list_nat @ A3 )
            = ( finite_card_list_nat @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9048_card__subset__eq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( ( finite_card_nat @ A3 )
            = ( finite_card_nat @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9049_card__subset__eq,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( ( finite4902975817058060853nteger @ A3 )
            = ( finite4902975817058060853nteger @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9050_card__subset__eq,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
       => ( ( ( finite711546835091564841at_nat @ A3 )
            = ( finite711546835091564841at_nat @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9051_card__subset__eq,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ( ( finite_card_int @ A3 )
            = ( finite_card_int @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_9052_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_o,R3: $o > $o > $o] :
      ( ( finite_finite_o @ B3 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A3 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B4: $o] :
              ( ( member_o @ A13 @ A3 )
             => ( ( member_o @ A24 @ A3 )
               => ( ( member_o @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9053_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_int,R3: int > $o > $o] :
      ( ( finite_finite_o @ B3 )
     => ( ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B4: $o] :
              ( ( member_int @ A13 @ A3 )
             => ( ( member_int @ A24 @ A3 )
               => ( ( member_o @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_o @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9054_card__le__if__inj__on__rel,axiom,
    ! [B3: set_o,A3: set_nat,R3: nat > $o > $o] :
      ( ( finite_finite_o @ B3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B4: $o] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member_o @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_o @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9055_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_o,R3: $o > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A3 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B4: nat] :
              ( ( member_o @ A13 @ A3 )
             => ( ( member_o @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9056_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_int,R3: int > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B4: nat] :
              ( ( member_int @ A13 @ A3 )
             => ( ( member_int @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9057_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_nat,R3: nat > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B4: nat] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9058_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_o,R3: $o > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A3 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B4: int] :
              ( ( member_o @ A13 @ A3 )
             => ( ( member_o @ A24 @ A3 )
               => ( ( member_int @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9059_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_int,R3: int > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B4: int] :
              ( ( member_int @ A13 @ A3 )
             => ( ( member_int @ A24 @ A3 )
               => ( ( member_int @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9060_card__le__if__inj__on__rel,axiom,
    ! [B3: set_int,A3: set_nat,R3: nat > int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B4: int] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member_int @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_int @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9061_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Code_integer,A3: set_o,R3: $o > code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ! [A4: $o] :
            ( ( member_o @ A4 @ A3 )
           => ? [B11: code_integer] :
                ( ( member_Code_integer @ B11 @ B3 )
                & ( R3 @ A4 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B4: code_integer] :
              ( ( member_o @ A13 @ A3 )
             => ( ( member_o @ A24 @ A3 )
               => ( ( member_Code_integer @ B4 @ B3 )
                 => ( ( R3 @ A13 @ B4 )
                   => ( ( R3 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite4902975817058060853nteger @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9062_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_9063_sum__multicount__gen,axiom,
    ! [S2: set_o,T5: set_o,R: $o > $o > $o,K: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T5 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9064_sum__multicount__gen,axiom,
    ! [S2: set_o,T5: set_int,R: $o > int > $o,K: int > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9065_sum__multicount__gen,axiom,
    ! [S2: set_o,T5: set_Code_integer,R: $o > code_integer > $o,K: code_integer > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J3: code_integer] :
                        ( ( member_Code_integer @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9066_sum__multicount__gen,axiom,
    ! [S2: set_int,T5: set_o,R: int > $o > $o,K: $o > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9067_sum__multicount__gen,axiom,
    ! [S2: set_int,T5: set_int,R: int > int > $o,K: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9068_sum__multicount__gen,axiom,
    ! [S2: set_int,T5: set_Code_integer,R: int > code_integer > $o,K: code_integer > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J3: code_integer] :
                        ( ( member_Code_integer @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9069_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T5: set_o,R: code_integer > $o > $o,K: $o > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T5 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I4: code_integer] :
                        ( ( member_Code_integer @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I4: code_integer] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9070_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T5: set_int,R: code_integer > int > $o,K: int > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I4: code_integer] :
                        ( ( member_Code_integer @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I4: code_integer] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9071_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T5: set_Code_integer,R: code_integer > code_integer > $o,K: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite6017078050557962740nteger @ T5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T5 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I4: code_integer] :
                        ( ( member_Code_integer @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I4: code_integer] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J3: code_integer] :
                        ( ( member_Code_integer @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9072_sum__multicount__gen,axiom,
    ! [S2: set_o,T5: set_nat,R: $o > nat > $o,K: nat > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X3 ) ) ) )
                = ( K @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T5 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T5 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9073_card__eq__sum,axiom,
    ( finite_card_set_nat
    = ( groups8294997508430121362at_nat
      @ ^ [X4: set_nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9074_card__eq__sum,axiom,
    ( finite_card_list_nat
    = ( groups4396056296759096172at_nat
      @ ^ [X4: list_nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9075_card__eq__sum,axiom,
    ( finite_card_nat
    = ( groups3542108847815614940at_nat
      @ ^ [X4: nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9076_card__eq__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9077_card__eq__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9078_card__eq__0__iff,axiom,
    ! [A3: set_Code_integer] :
      ( ( ( finite4902975817058060853nteger @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bo3990330152332043303nteger )
        | ~ ( finite6017078050557962740nteger @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9079_card__eq__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9080_card__eq__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9081_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9082_card__eq__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_9083_card__ge__0__finite,axiom,
    ! [A3: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9084_card__ge__0__finite,axiom,
    ! [A3: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
     => ( finite8100373058378681591st_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9085_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
     => ( finite_finite_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9086_card__ge__0__finite,axiom,
    ! [A3: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
     => ( finite_finite_int @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9087_card__ge__0__finite,axiom,
    ! [A3: set_Code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A3 ) )
     => ( finite6017078050557962740nteger @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9088_card__ge__0__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
     => ( finite6177210948735845034at_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_9089_card__image__le,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F @ A3 ) ) @ ( finite_card_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9090_card__image__le,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( finite_card_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9091_card__image__le,axiom,
    ! [A3: set_int,F: int > int] :
      ( ( finite_finite_int @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_int @ F @ A3 ) ) @ ( finite_card_int @ A3 ) ) ) ).

% card_image_le
thf(fact_9092_card__image__le,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_nat @ F @ A3 ) ) @ ( finite_card_int @ A3 ) ) ) ).

% card_image_le
thf(fact_9093_card__image__le,axiom,
    ! [A3: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_951025933927791156er_nat @ F @ A3 ) ) @ ( finite4902975817058060853nteger @ A3 ) ) ) ).

% card_image_le
thf(fact_9094_card__image__le,axiom,
    ! [A3: set_set_nat,F: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F @ A3 ) ) @ ( finite_card_set_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9095_card__image__le,axiom,
    ! [A3: set_list_nat,F: list_nat > nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_list_nat_nat @ F @ A3 ) ) @ ( finite_card_list_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9096_card__image__le,axiom,
    ! [A3: set_nat,F: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( finite_card_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9097_card__image__le,axiom,
    ! [A3: set_nat,F: nat > list_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( image_nat_list_nat @ F @ A3 ) ) @ ( finite_card_nat @ A3 ) ) ) ).

% card_image_le
thf(fact_9098_card__image__le,axiom,
    ! [A3: set_int,F: int > set_nat] :
      ( ( finite_finite_int @ A3 )
     => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( finite_card_int @ A3 ) ) ) ).

% card_image_le
thf(fact_9099_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_set_nat,C3: nat] :
      ( ! [G4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G4 @ F4 )
         => ( ( finite1152437895449049373et_nat @ G4 )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G4 ) @ C3 ) ) )
     => ( ( finite1152437895449049373et_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9100_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_list_nat,C3: nat] :
      ( ! [G4: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G4 @ F4 )
         => ( ( finite8100373058378681591st_nat @ G4 )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G4 ) @ C3 ) ) )
     => ( ( finite8100373058378681591st_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9101_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_nat,C3: nat] :
      ( ! [G4: set_nat] :
          ( ( ord_less_eq_set_nat @ G4 @ F4 )
         => ( ( finite_finite_nat @ G4 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C3 ) ) )
     => ( ( finite_finite_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9102_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_Code_integer,C3: nat] :
      ( ! [G4: set_Code_integer] :
          ( ( ord_le7084787975880047091nteger @ G4 @ F4 )
         => ( ( finite6017078050557962740nteger @ G4 )
           => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ G4 ) @ C3 ) ) )
     => ( ( finite6017078050557962740nteger @ F4 )
        & ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9103_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ! [G4: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ G4 @ F4 )
         => ( ( finite6177210948735845034at_nat @ G4 )
           => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G4 ) @ C3 ) ) )
     => ( ( finite6177210948735845034at_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9104_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_int,C3: nat] :
      ( ! [G4: set_int] :
          ( ( ord_less_eq_set_int @ G4 @ F4 )
         => ( ( finite_finite_int @ G4 )
           => ( ord_less_eq_nat @ ( finite_card_int @ G4 ) @ C3 ) ) )
     => ( ( finite_finite_int @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F4 ) @ C3 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9105_card__seteq,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B3 ) @ ( finite_card_set_nat @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9106_card__seteq,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B3 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B3 ) @ ( finite_card_list_nat @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9107_card__seteq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9108_card__seteq,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ B3 ) @ ( finite4902975817058060853nteger @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9109_card__seteq,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B3 ) @ ( finite711546835091564841at_nat @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9110_card__seteq,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B3 ) @ ( finite_card_int @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_9111_card__mono,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_9112_card__mono,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B3 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_9113_card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_9114_card__mono,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le7084787975880047091nteger @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A3 ) @ ( finite4902975817058060853nteger @ B3 ) ) ) ) ).

% card_mono
thf(fact_9115_card__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_9116_card__mono,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B3 ) ) ) ) ).

% card_mono
thf(fact_9117_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ S ) )
     => ~ ! [T6: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T6 @ S )
           => ( ( ( finite_card_set_nat @ T6 )
                = N2 )
             => ~ ( finite1152437895449049373et_nat @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9118_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_list_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ S ) )
     => ~ ! [T6: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T6 @ S )
           => ( ( ( finite_card_list_nat @ T6 )
                = N2 )
             => ~ ( finite8100373058378681591st_nat @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9119_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
     => ~ ! [T6: set_nat] :
            ( ( ord_less_eq_set_nat @ T6 @ S )
           => ( ( ( finite_card_nat @ T6 )
                = N2 )
             => ~ ( finite_finite_nat @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9120_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Code_integer] :
      ( ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ S ) )
     => ~ ! [T6: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ T6 @ S )
           => ( ( ( finite4902975817058060853nteger @ T6 )
                = N2 )
             => ~ ( finite6017078050557962740nteger @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9121_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite711546835091564841at_nat @ S ) )
     => ~ ! [T6: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T6 @ S )
           => ( ( ( finite711546835091564841at_nat @ T6 )
                = N2 )
             => ~ ( finite6177210948735845034at_nat @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9122_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_int] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S ) )
     => ~ ! [T6: set_int] :
            ( ( ord_less_eq_set_int @ T6 @ S )
           => ( ( ( finite_card_int @ T6 )
                = N2 )
             => ~ ( finite_finite_int @ T6 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9123_card__less__sym__Diff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B3 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B3 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9124_card__less__sym__Diff,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B3 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B3 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9125_card__less__sym__Diff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B3 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9126_card__less__sym__Diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ A3 ) @ ( finite4902975817058060853nteger @ B3 ) )
         => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9127_card__less__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9128_card__less__sym__Diff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9129_card__le__sym__Diff,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B3 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9130_card__le__sym__Diff,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B3 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9131_card__le__sym__Diff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B3 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9132_card__le__sym__Diff,axiom,
    ! [A3: set_Code_integer,B3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A3 )
     => ( ( finite6017078050557962740nteger @ B3 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A3 ) @ ( finite4902975817058060853nteger @ B3 ) )
         => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A3 @ B3 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9133_card__le__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9134_card__le__sym__Diff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9135_psubset__card__mono,axiom,
    ! [B3: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_less_set_set_nat @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9136_psubset__card__mono,axiom,
    ! [B3: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B3 )
     => ( ( ord_le1190675801316882794st_nat @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9137_psubset__card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_set_nat @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9138_psubset__card__mono,axiom,
    ! [B3: set_int,A3: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_set_int @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9139_psubset__card__mono,axiom,
    ! [B3: set_Code_integer,A3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B3 )
     => ( ( ord_le1307284697595431911nteger @ A3 @ B3 )
       => ( ord_less_nat @ ( finite4902975817058060853nteger @ A3 ) @ ( finite4902975817058060853nteger @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9140_psubset__card__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( ord_le7866589430770878221at_nat @ A3 @ B3 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_9141_card__Un__le,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9142_card__Un__le,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9143_card__Un__le,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9144_card__Un__le,axiom,
    ! [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : ( ord_less_eq_nat @ ( finite7007076676225009423at_nat @ ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite7007076676225009423at_nat @ A3 ) @ ( finite7007076676225009423at_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9145_card__Un__le,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9146_card__Un__le,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ).

% card_Un_le
thf(fact_9147_card__less__Suc2,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M6 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M6 )
                & ( ord_less_nat @ K3 @ I2 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M6 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9148_card__less__Suc,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M6 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M6 )
                  & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M6 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9149_card__less,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M6 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M6 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9150_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
     => ( A3
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9151_sum__Suc,axiom,
    ! [F: set_nat > nat,A3: set_set_nat] :
      ( ( groups8294997508430121362at_nat
        @ ^ [X4: set_nat] : ( suc @ ( F @ X4 ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( finite_card_set_nat @ A3 ) ) ) ).

% sum_Suc
thf(fact_9152_sum__Suc,axiom,
    ! [F: list_nat > nat,A3: set_list_nat] :
      ( ( groups4396056296759096172at_nat
        @ ^ [X4: list_nat] : ( suc @ ( F @ X4 ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A3 ) @ ( finite_card_list_nat @ A3 ) ) ) ).

% sum_Suc
thf(fact_9153_sum__Suc,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( finite_card_nat @ A3 ) ) ) ).

% sum_Suc
thf(fact_9154_sum__multicount,axiom,
    ! [S: set_o,T: set_o,R: $o > $o > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_o @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_o @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9155_sum__multicount,axiom,
    ! [S: set_o,T: set_nat,R: $o > nat > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9156_sum__multicount,axiom,
    ! [S: set_o,T: set_int,R: $o > int > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9157_sum__multicount,axiom,
    ! [S: set_o,T: set_Code_integer,R: $o > code_integer > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J3: code_integer] :
                        ( ( member_Code_integer @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite4902975817058060853nteger @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9158_sum__multicount,axiom,
    ! [S: set_int,T: set_o,R: int > $o > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_o @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_o @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9159_sum__multicount,axiom,
    ! [S: set_int,T: set_nat,R: int > nat > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9160_sum__multicount,axiom,
    ! [S: set_int,T: set_int,R: int > int > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9161_sum__multicount,axiom,
    ! [S: set_int,T: set_Code_integer,R: int > code_integer > $o,K: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I4: int] :
                        ( ( member_int @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I4: int] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J3: code_integer] :
                        ( ( member_Code_integer @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite4902975817058060853nteger @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9162_sum__multicount,axiom,
    ! [S: set_Code_integer,T: set_o,R: code_integer > $o > $o,K: nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_o @ T )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I4: code_integer] :
                        ( ( member_Code_integer @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I4: code_integer] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_o @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9163_sum__multicount,axiom,
    ! [S: set_Code_integer,T: set_nat,R: code_integer > nat > $o,K: nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I4: code_integer] :
                        ( ( member_Code_integer @ I4 @ S )
                        & ( R @ I4 @ X3 ) ) ) )
                = K ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I4: code_integer] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9164_card__gt__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_set_nat )
        & ( finite1152437895449049373et_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9165_card__gt__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_list_nat )
        & ( finite8100373058378681591st_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9166_card__gt__0__iff,axiom,
    ! [A3: set_Code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A3 ) )
      = ( ( A3 != bot_bo3990330152332043303nteger )
        & ( finite6017078050557962740nteger @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9167_card__gt__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
      = ( ( A3 != bot_bo2099793752762293965at_nat )
        & ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9168_card__gt__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
      = ( ( A3 != bot_bot_set_o )
        & ( finite_finite_o @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9169_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_nat )
        & ( finite_finite_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9170_card__gt__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
      = ( ( A3 != bot_bot_set_int )
        & ( finite_finite_int @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_9171_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: code_integer,F: nat > code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_le3102999989581377725nteger @ K5 @ ( F @ I3 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A3 ) ) @ K5 ) @ ( groups7501900531339628137nteger @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9172_sum__bounded__below,axiom,
    ! [A3: set_o,K5: rat,F: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ord_less_eq_rat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K5 ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9173_sum__bounded__below,axiom,
    ! [A3: set_int,K5: rat,F: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_rat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K5 ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9174_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: rat,F: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_rat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K5 ) @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9175_sum__bounded__below,axiom,
    ! [A3: set_o,K5: nat,F: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ord_less_eq_nat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A3 ) ) @ K5 ) @ ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9176_sum__bounded__below,axiom,
    ! [A3: set_int,K5: nat,F: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_nat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A3 ) ) @ K5 ) @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9177_sum__bounded__below,axiom,
    ! [A3: set_o,K5: int,F: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A3 )
         => ( ord_less_eq_int @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A3 ) ) @ K5 ) @ ( groups8505340233167759370_o_int @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9178_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: int,F: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_int @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A3 ) ) @ K5 ) @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9179_sum__bounded__below,axiom,
    ! [A3: set_nat,K5: nat,F: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_nat @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A3 ) ) @ K5 ) @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9180_sum__bounded__below,axiom,
    ! [A3: set_int,K5: int,F: int > int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_int @ K5 @ ( F @ I3 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_int @ A3 ) ) @ K5 ) @ ( groups4538972089207619220nt_int @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_9181_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9182_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_9183_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).

% integer_of_num(2)
thf(fact_9184_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9185_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9186_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9187_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_9188_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu2: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9189_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu2: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9190_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% int_of_integer_less_iff
thf(fact_9191_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_9192_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9193_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9194_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9195_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_9196_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9197_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R4: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R4 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9198_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9199_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_9200_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_9201_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_9202_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_9203_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_9204_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9205_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( 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 @ L2 @ S6 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( 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 @ L2 ) @ S6 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9206_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9207_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9208_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_9209_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9210_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N5 @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K @ M4 )
                = ( bit_se1146084159140164899it_int @ K @ N5 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N5 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N5 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9211_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_9212_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [R2: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
             => ( member_nat @ ( R2 @ N9 ) @ S ) ) ) ) ).

% finite_enumerate
thf(fact_9213_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9214_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9215_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9216_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9217_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_9218_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9219_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9220_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q6: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q6 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q6 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9221_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_9222_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9223_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( 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 @ L2 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9224_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_9225_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9226_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9227_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9228_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_9229_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_9230_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_9231_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_9232_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9233_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9234_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9235_mod__true,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H )
      = ( in_range @ H ) ) ).

% mod_true
thf(fact_9236_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9237_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9238_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9239_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9240_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9241_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9242_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_9243_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q6: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some_num @ Q6 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = ( numeral_numeral_int @ Q6 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9244_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9245_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_9246_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_9247_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_9248_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_9249_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9250_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9251_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H ) ) ).

% mod_star_trueI
thf(fact_9252_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_9253_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_9254_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_9255_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9256_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9257_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_9258_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9259_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_9260_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_9261_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ 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 ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9262_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9263_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_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ 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_9264_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9265_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9266_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_9267_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9268_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9269_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9270_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9271_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_9272_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_9273_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_9274_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9275_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9276_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9277_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9278_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_nat @ 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_9279_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] :
                  ( Xa
                  = ( bit0 @ N5 ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] :
                    ( Xa
                    = ( bit1 @ N5 ) )
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N5: num] :
                        ( ( Xa
                          = ( bit0 @ N5 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit1 @ N5 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N5: num] :
                              ( ( Xa
                                = ( bit0 @ N5 ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit1 @ N5 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9280_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa
                    = ( bit0 @ N5 ) )
                 => ( ( Y
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N5 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa
                      = ( bit1 @ N5 ) )
                   => ( ( Y = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N5 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit0 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit0 @ N5 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N5: num] :
                            ( ( Xa
                              = ( bit1 @ N5 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit0 @ N5 ) )
                               => ( ( Y
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                      @ ( bit_and_not_num @ M5 @ N5 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N5: num] :
                                  ( ( Xa
                                    = ( bit1 @ N5 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_9281_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] :
                  ( Xa
                  = ( bit0 @ N5 ) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] :
                    ( Xa
                    = ( bit1 @ N5 ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M5: num] :
                    ( X
                    = ( bit0 @ M5 ) )
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N5: num] :
                        ( ( Xa
                          = ( bit0 @ N5 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit1 @ N5 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
                 => ( ( ? [M5: num] :
                          ( X
                          = ( bit1 @ M5 ) )
                     => ( ( Xa = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N5: num] :
                              ( ( Xa
                                = ( bit0 @ N5 ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit1 @ N5 ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9282_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N5: num] :
                ( ( Xa
                  = ( bit0 @ N5 ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N5 ) ) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa
                    = ( bit1 @ N5 ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N5 ) ) ) ) )
           => ( ! [M5: num] :
                  ( ( X
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N5: num] :
                        ( ( Xa
                          = ( bit0 @ N5 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit1 @ N5 ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N5: num] :
                              ( ( Xa
                                = ( bit0 @ N5 ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit1 @ N5 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9283_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9284_and__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9285_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9286_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ ( bit1 @ N2 ) ) ) ).

% xor_num.simps(2)
thf(fact_9287_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ ( bit0 @ N2 ) ) ) ).

% xor_num.simps(3)
thf(fact_9288_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_9289_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_9290_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(9)
thf(fact_9291_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9292_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9293_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa
                    = ( bit0 @ N5 ) )
                 => ( ( Y = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N5 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa
                      = ( bit1 @ N5 ) )
                   => ( ( Y
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N5 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit0 @ N5 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N5: num] :
                            ( ( Xa
                              = ( bit1 @ N5 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit0 @ N5 ) )
                               => ( ( Y
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N5: num] :
                                  ( ( Xa
                                    = ( bit1 @ N5 ) )
                                 => ( ( Y
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_9294_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa
                    = ( bit0 @ N5 ) )
                 => ( ( Y
                      = ( some_num @ ( bit1 @ N5 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N5 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa
                      = ( bit1 @ N5 ) )
                   => ( ( Y
                        = ( some_num @ ( bit0 @ N5 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N5 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit1 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N5: num] :
                          ( ( Xa
                            = ( bit0 @ N5 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N5: num] :
                            ( ( Xa
                              = ( bit1 @ N5 ) )
                           => ( ( Y
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N5: num] :
                                ( ( Xa
                                  = ( bit0 @ N5 ) )
                               => ( ( Y
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N5: num] :
                                  ( ( Xa
                                    = ( bit1 @ N5 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_9295_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
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) )
          @ Xa
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9296_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_9297_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9298_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9299_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9300_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9301_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
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) )
        @ Xa
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9302_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) )
        @ Xa
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9303_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
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) )
          @ Xa
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9304_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
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) )
          @ Xa
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9305_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y4: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9306_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y4: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9307_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) ) ) ) ).

% uminus_int_def
thf(fact_9308_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M3: nat,N: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9309_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9310_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9311_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9312_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( ( finite_finite_nat @ M6 )
     => ( ( gcd_Gcd_nat @ M6 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M6 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9313_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9314_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( ( finite_finite_nat @ M6 )
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M6 )
         => ( ( gcd_Gcd_nat @ M6 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M3: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M3 ) )
                  @ M6 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9315_Gcd__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_abs_eq
thf(fact_9316_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_nat
        @ ( image_int_nat
          @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
          @ K5 ) )
      = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).

% Gcd_nat_abs_eq
thf(fact_9317_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9318_Gcd__int__eq,axiom,
    ! [N7: set_nat] :
      ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N7 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N7 ) ) ) ).

% Gcd_int_eq
thf(fact_9319_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_9320_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X11: set_nat] : ( if_nat @ ( X11 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X11 ) ) ) ) ).

% Sup_nat_def
thf(fact_9321_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9322_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9323_Gcd__int__def,axiom,
    ( gcd_Gcd_int
    = ( ^ [K6: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K6 ) ) ) ) ) ).

% Gcd_int_def
thf(fact_9324_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_9325_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9326_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9327_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9328_mono__times__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).

% mono_times_nat
thf(fact_9329_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9330_binomial__def,axiom,
    ( binomial
    = ( ^ [N: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K6: set_nat] :
                ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
                & ( ( finite_card_nat @ K6 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9331_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9332_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
     => ( ( ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9333_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_9334_sort__upto,axiom,
    ! [I2: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X4: int] : X4
        @ ( upto @ I2 @ J ) )
      = ( upto @ I2 @ J ) ) ).

% sort_upto
thf(fact_9335_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( ( upto @ I2 @ J )
        = nil_int )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil
thf(fact_9336_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int
        = ( upto @ I2 @ J ) )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil2
thf(fact_9337_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( ( ord_less_int @ J @ I2 )
     => ( ( upto @ I2 @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9338_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
        = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9339_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9340_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9341_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9342_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9343_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_9344_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J ) ) ).

% sorted_wrt_upto
thf(fact_9345_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9346_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9347_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_9348_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9349_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9350_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9351_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9352_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N7 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N7 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N7 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9353_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N7 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).

% card_length_sum_list
thf(fact_9354_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X4: nat] : X4
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_9355_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_9356_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt
thf(fact_9357_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( upt @ I2 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9358_upt__merge,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_eq_nat @ J @ K ) )
     => ( ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I2 @ K ) ) ) ).

% upt_merge
thf(fact_9359_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( upt @ I2 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9360_take__upt,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I2 @ N2 ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_9361_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9362_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( last_nat @ ( upt @ I2 @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_9363_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_9364_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_9365_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_9366_upt__eq__append__conv,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( upt @ I2 @ J )
          = ( append_nat @ Xs @ Ys3 ) )
        = ( ? [K3: nat] :
              ( ( ord_less_eq_nat @ I2 @ K3 )
              & ( ord_less_eq_nat @ K3 @ J )
              & ( ( upt @ I2 @ K3 )
                = Xs )
              & ( ( upt @ K3 @ J )
                = Ys3 ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_9367_upt__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I2 ) @ ( upt @ I2 @ J ) )
        = ( upt @ zero_zero_nat @ J ) ) ) ).

% upt_append
thf(fact_9368_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9369_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_9370_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9371_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9372_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9373_butlast__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( butlast_nat @ ( upt @ M @ N2 ) )
      = ( upt @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_9374_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_9375_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9376_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_9377_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
      ( ( ( upt @ L @ H )
        = ( append_nat @ Is1 @ ( cons_nat @ I2 @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L @ I2 ) )
        & ( Is2
          = ( upt @ ( suc @ I2 ) @ H ) )
        & ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_nat @ I2 @ H ) ) ) ).

% upt_eq_lel_conv
thf(fact_9378_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9379_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq_nat @ U @ I3 )
              & ( ord_less_nat @ I3 @ U3 ) )
           => ~ ( P @ I3 ) )
       => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_9380_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_9381_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_9382_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y4: nat] : ( if_Pro3027730157355071871nt_int @ ( Y4 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y4 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9383_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_9384_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_9385_mod__pure,axiom,
    ! [B: $o,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B ) @ H )
      = ( ( ( produc8586169260539613262et_nat @ H )
          = bot_bot_set_nat )
        & B ) ) ).

% mod_pure
thf(fact_9386_mod__emp,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H )
      = ( ( produc8586169260539613262et_nat @ H )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_9387_mod__star__trueE_H,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H5 )
              = ( produc1824681642469235216et_nat @ H ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H5 ) @ ( produc8586169260539613262et_nat @ H ) )
             => ~ ( rep_assn @ P @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_9388_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_9389_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% less_eq
thf(fact_9390_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_9391_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9392_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9393_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_9394_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_9395_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_9396_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q6 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q6 ) ) ) ).

% nat_mult_min_right
thf(fact_9397_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q6 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q6 ) @ ( times_times_nat @ N2 @ Q6 ) ) ) ).

% nat_mult_min_left
thf(fact_9398_min__diff,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N2 @ I2 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I2 ) ) ).

% min_diff
thf(fact_9399_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9400_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ M7 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_9401_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ N2 @ M7 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_9402_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9403_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9404_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ) ).

% less_int_def
thf(fact_9405_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9406_less__eq__integer__def,axiom,
    ( ord_le3102999989581377725nteger
    = ( map_fu2496120808607739376eger_o @ code_int_of_integer @ ( map_fu6957801986076833569nt_o_o @ code_int_of_integer @ id_o ) @ ord_less_eq_int ) ) ).

% less_eq_integer_def
thf(fact_9407_less__integer__def,axiom,
    ( ord_le6747313008572928689nteger
    = ( map_fu2496120808607739376eger_o @ code_int_of_integer @ ( map_fu6957801986076833569nt_o_o @ code_int_of_integer @ id_o ) @ ord_less_int ) ) ).

% less_integer_def
thf(fact_9408_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_9409_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_9410_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_9411_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_9412_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_9413_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_9414_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_9415_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q6 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q6 ) @ ( times_times_nat @ N2 @ Q6 ) ) ) ).

% nat_mult_max_left
thf(fact_9416_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q6 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q6 ) ) ) ).

% nat_mult_max_right
thf(fact_9417_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q6 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q6 ) ) ) ).

% nat_add_max_right
thf(fact_9418_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q6 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q6 ) @ ( plus_plus_nat @ N2 @ Q6 ) ) ) ).

% nat_add_max_left
thf(fact_9419_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_9420_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ N2 @ M7 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9421_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ M7 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9422_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9423_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9424_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_9425_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N5: nat] :
          ( ( member_nat @ N5 @ N7 )
         => ( ord_less_eq_nat @ K @ N5 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_9426_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X4: rat] :
          ( the_int
          @ ^ [Z6: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X4 )
              & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9427_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9428_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_rat_def
thf(fact_9429_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9430_normalize__negative,axiom,
    ! [Q6: int,P3: int] :
      ( ( ord_less_int @ Q6 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P3 @ Q6 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P3 ) @ ( uminus_uminus_int @ Q6 ) ) ) ) ) ).

% normalize_negative
thf(fact_9431_normalize__denom__zero,axiom,
    ! [P3: int] :
      ( ( normalize @ ( product_Pair_int_int @ P3 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9432_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P3: int,Q6: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% normalize_denom_pos
thf(fact_9433_normalize__crossproduct,axiom,
    ! [Q6: int,S2: int,P3: int,R3: int] :
      ( ( Q6 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P3 @ Q6 ) )
            = ( normalize @ ( product_Pair_int_int @ R3 @ S2 ) ) )
         => ( ( times_times_int @ P3 @ S2 )
            = ( times_times_int @ R3 @ Q6 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9434_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_9435_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_9436_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
      = ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9437_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9438_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9439_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9440_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9441_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9442_quotient__of__div,axiom,
    ! [R3: rat,N2: int,D2: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ N2 @ D2 ) )
     => ( R3
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_9443_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).

% gcd_le1_int
thf(fact_9444_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).

% gcd_le2_int
thf(fact_9445_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9446_gcd__unique__int,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D2 )
        & ( dvd_dvd_int @ D2 @ A )
        & ( dvd_dvd_int @ D2 @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D2 ) ) )
      = ( D2
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_9447_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ Y )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_non_0_int
thf(fact_9448_quotient__of__denom__pos,axiom,
    ! [R3: rat,P3: int,Q6: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% quotient_of_denom_pos
thf(fact_9449_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R3 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9450_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D5: int] :
                ( ( dvd_dvd_int @ D5 @ M )
                & ( dvd_dvd_int @ D5 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9451_rat__uminus__code,axiom,
    ! [P3: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_uminus_code
thf(fact_9452_rat__times__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( times_times_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B5 ) @ ( times_times_int @ C4 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_times_code
thf(fact_9453_rat__divide__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C4 @ B5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_divide_code
thf(fact_9454_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P7: rat,Q7: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C4: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B5: int,D5: int] : ( ord_less_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C4 @ B5 ) )
              @ ( quotient_of @ Q7 ) )
          @ ( quotient_of @ P7 ) ) ) ) ).

% rat_less_code
thf(fact_9455_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P7: rat,Q7: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C4: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B5: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C4 @ B5 ) )
              @ ( quotient_of @ Q7 ) )
          @ ( quotient_of @ P7 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9456_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P7: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P7 ) ) ) ) ).

% rat_floor_code
thf(fact_9457_rat__abs__code,axiom,
    ! [P3: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_abs_code
thf(fact_9458_rat__plus__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_plus_code
thf(fact_9459_rat__minus__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_minus_code
thf(fact_9460_rat__sgn__code,axiom,
    ! [P3: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P3 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P3 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9461_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9462_rat__inverse__code,axiom,
    ! [P3: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B5 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_inverse_code
thf(fact_9463_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9464_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9465_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9466_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9467_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9468_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member_nat @ A4 @ A3 )
         => ( ( member_nat @ B4 @ A3 )
           => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A3 ) ) )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).

% Gcd_in
thf(fact_9469_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9470_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M3: nat,N: nat] :
        ( ( dvd_dvd_nat @ M3 @ N )
        & ( M3 != N ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9471_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M )
                & ( dvd_dvd_nat @ D5 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9472_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_9473_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9474_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9475_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9476_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_9477_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9478_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9479_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_9480_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9481_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9482_less__rat,axiom,
    ! [B: int,D2: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D2 ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% less_rat
thf(fact_9483_le__rat,axiom,
    ! [B: int,D2: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D2 ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% le_rat
thf(fact_9484_quotient__of__eq,axiom,
    ! [A: int,B: int,P3: int,Q6: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ( fract @ P3 @ Q6 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_9485_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q6: rat] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( P @ ( fract @ A4 @ B4 ) ) )
     => ( P @ Q6 ) ) ).

% Rat_induct_pos
thf(fact_9486_normalize__eq,axiom,
    ! [A: int,B: int,P3: int,Q6: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ( fract @ P3 @ Q6 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_9487_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_9488_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9489_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_9490_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_9491_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9492_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_9493_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9494_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9495_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_9496_positive__rat,axiom,
    ! [A: int,B: int] :
      ( ( positive @ ( fract @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% positive_rat
thf(fact_9497_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] : ( positive @ ( minus_minus_rat @ Y4 @ X4 ) ) ) ) ).

% less_rat_def
thf(fact_9498_positive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).

% positive_def
thf(fact_9499_positive_Orep__eq,axiom,
    ( positive
    = ( ^ [X4: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X4 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).

% positive.rep_eq
thf(fact_9500_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ) ).

% plus_rat_def
thf(fact_9501_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X4: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X4 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_9502_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_9503_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X4: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X4 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_9504_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_9505_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( B5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B5 ) ) ) ) ).

% Fract_def
thf(fact_9506_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ) ).

% uminus_rat_def
thf(fact_9507_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ) ).

% times_rat_def
thf(fact_9508_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_9509_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_9510_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_9511_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X4: product_prod_int_int,Y4: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X4 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y4 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).

% ratrel_def
thf(fact_9512_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_9513_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_9514_positive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( positive @ ( abs_Rat @ X ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% positive.abs_eq
thf(fact_9515_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_9516_cr__rat__def,axiom,
    ( cr_rat
    = ( ^ [X4: product_prod_int_int,Y4: rat] :
          ( ( ratrel @ X4 @ X4 )
          & ( ( abs_Rat @ X4 )
            = Y4 ) ) ) ) ).

% cr_rat_def
thf(fact_9517_Code__Numeral_Odup__def,axiom,
    ( code_dup
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
      @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ) ).

% Code_Numeral.dup_def
thf(fact_9518_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X4: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X4
                = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).

% quotient_of_def
thf(fact_9519_normalize__stable,axiom,
    ! [Q6: int,P3: int] :
      ( ( ord_less_int @ zero_zero_int @ Q6 )
     => ( ( algebr932160517623751201me_int @ P3 @ Q6 )
       => ( ( normalize @ ( product_Pair_int_int @ P3 @ Q6 ) )
          = ( product_Pair_int_int @ P3 @ Q6 ) ) ) ) ).

% normalize_stable
thf(fact_9520_quotient__of__coprime,axiom,
    ! [R3: rat,P3: int,Q6: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( algebr932160517623751201me_int @ P3 @ Q6 ) ) ).

% quotient_of_coprime
thf(fact_9521_normalize__coprime,axiom,
    ! [R3: product_prod_int_int,P3: int,Q6: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( algebr932160517623751201me_int @ P3 @ Q6 ) ) ).

% normalize_coprime
thf(fact_9522_Rat__induct,axiom,
    ! [P: rat > $o,Q6: rat] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ( algebr932160517623751201me_int @ A4 @ B4 )
           => ( P @ ( fract @ A4 @ B4 ) ) ) )
     => ( P @ Q6 ) ) ).

% Rat_induct
thf(fact_9523_Rat__cases,axiom,
    ! [Q6: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q6
            = ( fract @ A4 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) ).

% Rat_cases
thf(fact_9524_Rat__cases__nonzero,axiom,
    ! [Q6: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q6
            = ( fract @ A4 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ( ( A4 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) )
     => ( Q6 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_9525_quotient__of__unique,axiom,
    ! [R3: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R3
        = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
      & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
      & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
      & ! [Y6: product_prod_int_int] :
          ( ( ( R3
              = ( fract @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y6 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
         => ( Y6 = X3 ) ) ) ).

% quotient_of_unique
thf(fact_9526_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).

% coprime_diff_one_left_nat
thf(fact_9527_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_9528_positive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).

% positive.rsp
thf(fact_9529_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y5: num,Z3: num] : Y5 = Z3
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y5: num,Z3: num] : Y5 = Z3
      @ ^ [Y5: int,Z3: int] : Y5 = Z3 )
    @ ^ [M3: num,N: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) )
    @ ^ [M3: num,N: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ).

% sub.rsp
thf(fact_9530_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ratrel )
    @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( B5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B5 ) )
    @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( B5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B5 ) ) ) ).

% Fract.rsp
thf(fact_9531_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).

% dup.rsp
thf(fact_9532_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_9533_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9534_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9535_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_9536_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ).

% times_rat.rsp
thf(fact_9537_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ).

% plus_rat.rsp
thf(fact_9538_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ).

% uminus_rat.rsp
thf(fact_9539_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_9540_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_9541_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_9542_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_9543_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ pcr_rat )
    @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( B5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B5 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_9544_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_9545_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_9546_positive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
    @ positive ) ).

% positive.transfer
thf(fact_9547_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_9548_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_9549_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_9550_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ pcr_int
    @ ^ [N: nat] : ( product_Pair_nat_nat @ N @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9551_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_9552_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_9553_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_9554_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_9555_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9556_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_9557_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_9558_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X11: set_int] :
          ( the_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ X11 )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ X11 )
                 => ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_9559_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_9560_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_9561_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_9562_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) ) ) ).

% uminus_int.rsp
thf(fact_9563_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_9564_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_9565_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] :
              ( ( plus_plus_nat @ X4 @ V2 )
              = ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).

% intrel_def
thf(fact_9566_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).

% less_int.rsp
thf(fact_9567_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9568_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_9569_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_9570_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9571_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9572_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9573_Field__natLeq__on,axiom,
    ! [N2: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ X4 @ N2 )
                & ( ord_less_nat @ Y4 @ N2 )
                & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) ).

% Field_natLeq_on
thf(fact_9574_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_9575_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_9576_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9577_le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ).

% le_enumerate
thf(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9579_finite__le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S ) )
       => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S @ N2 ) ) ) ) ).

% finite_le_enumerate
thf(fact_9580_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9581_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_9582_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_9583_Inf__nat__def,axiom,
    ( complete_Inf_Inf_nat
    = ( ^ [X11: set_nat] :
          ( ord_Least_nat
          @ ^ [N: nat] : ( member_nat @ N @ X11 ) ) ) ) ).

% Inf_nat_def
thf(fact_9584_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_9585_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_9586_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_9587_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9588_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_9589_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_9590_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9591_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_9592_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_9593_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_9594_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9595_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_9596_bij__betw__Suc,axiom,
    ! [M6: set_nat,N7: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M6 @ N7 )
      = ( ( image_nat_nat @ suc @ M6 )
        = N7 ) ) ).

% bij_betw_Suc
thf(fact_9597_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_9598_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( product_Rep_unit @ X )
        = ( product_Rep_unit @ Y ) )
      = ( X = Y ) ) ).

% Rep_unit_inject
thf(fact_9599_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_9600_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X3: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).

% Rep_unit_cases
thf(fact_9601_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_9602_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_9603_Rep__unit__inverse,axiom,
    ! [X: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
      = X ) ).

% Rep_unit_inverse
thf(fact_9604_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X
            = ( product_Abs_unit @ Y3 ) )
         => ~ ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_9605_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y3 ) ) )
     => ( P @ X ) ) ).

% Abs_unit_induct
thf(fact_9606_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X )
            = ( product_Abs_unit @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_9607_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_9608_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9609_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F4 ) ) ) ).

% le_sequentially
thf(fact_9610_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_9611_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N8 @ N )
           => ( P @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_9612_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_9613_natLeq__on__wo__rel,axiom,
    ! [N2: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y4 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_9614_UNIV__unit,axiom,
    ( top_to1996260823553986621t_unit
    = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).

% UNIV_unit
thf(fact_9615_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
thf(fact_9616_Unity__def,axiom,
    ( product_Unity
    = ( product_Abs_unit @ $true ) ) ).

% Unity_def
thf(fact_9617_sup__unit__def,axiom,
    ( sup_sup_Product_unit
    = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).

% sup_unit_def
thf(fact_9618_Inf__unit__def,axiom,
    ( comple2584293577114468500t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Inf_unit_def
thf(fact_9619_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
thf(fact_9620_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
thf(fact_9621_uminus__unit__def,axiom,
    ( uminus2952777764628376836t_unit
    = ( ^ [Uu3: product_unit] : product_Unity ) ) ).

% uminus_unit_def
thf(fact_9622_Sup__unit__def,axiom,
    ( comple4687483117567863418t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Sup_unit_def
thf(fact_9623_inf__unit__def,axiom,
    ( inf_inf_Product_unit
    = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).

% inf_unit_def
thf(fact_9624_wait__def,axiom,
    ( heap_Time_wait
    = ( ^ [N: nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H6 @ N ) ) ) ) ) ) ).

% wait_def
thf(fact_9625_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
thf(fact_9626_log_Osimps,axiom,
    ( log
    = ( ^ [B5: code_natural,I4: code_natural] :
          ( if_Code_natural
          @ ( ( ord_le1926595141338095240atural @ B5 @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ I4 @ B5 ) )
          @ one_one_Code_natural
          @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B5 @ ( divide5121882707175180666atural @ I4 @ B5 ) ) ) ) ) ) ).

% log.simps
thf(fact_9627_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( ( log @ X @ Xa )
        = Y )
     => ( ( ( ( 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 ) ) ) ) ) ) ) ).

% log.elims
thf(fact_9628_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_9629_minus__shift__def,axiom,
    ( minus_shift
    = ( ^ [R4: code_natural,K3: code_natural,L2: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K3 @ L2 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R4 @ K3 ) @ L2 ) @ ( minus_7197305767214868737atural @ K3 @ L2 ) ) ) ) ).

% minus_shift_def
thf(fact_9630_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc8731074985263844745atural] :
      ~ ! [F2: code_natural > option6357759511663192854e_term,D: code_natural,I3: code_natural] :
          ( X
         != ( produc2252593628808123835atural @ F2 @ ( produc3574140220909816553atural @ D @ I3 ) ) ) ).

% exhaustive_natural'.cases
thf(fact_9631_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc989692990947075319atural] :
      ~ ! [F2: produc4972180933644002618e_term > option6357759511663192854e_term,D: code_natural,I3: code_natural] :
          ( X
         != ( produc3831813291587773865atural @ F2 @ ( produc3574140220909816553atural @ D @ I3 ) ) ) ).

% full_exhaustive_natural'.cases
thf(fact_9632_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B4: code_natural,I3: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ B4 @ I3 ) ) ).

% log.cases
thf(fact_9633_next_Osimps,axiom,
    ! [V: code_natural,W2: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V @ W2 ) )
      = ( 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 @ V @ ( 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 @ V @ ( 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 @ W2 @ ( 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 @ W2 @ ( 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 @ V @ ( 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 @ V @ ( 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 @ W2 @ ( 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 @ W2 @ ( 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_9634_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S6: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V2: code_natural,W3: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V3: code_natural,W4: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W4 ) @ ( produc3574140220909816553atural @ V3 @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W3 ) ) )
              @ ( produc6591912806276919810atural @ ( next @ S6 ) ) )
          @ S6 ) ) ) ).

% split_seed_def
thf(fact_9635_Random_Orange__def,axiom,
    ( range
    = ( ^ [K3: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
            @ ^ [L2: code_natural] :
                ( produc5538323210962509403atural @ next
                @ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V2 @ ( times_2397367101498566445atural @ L2 @ ( 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 )
          @ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V2 @ K3 ) ) ) ) ) ).

% Random.range_def
thf(fact_9636_range,axiom,
    ! [K: code_natural,S2: produc7822875418678951345atural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ K )
     => ( ord_le5570908160329646204atural @ ( produc497848011034438852atural @ ( range @ K @ S2 ) ) @ K ) ) ).

% range
thf(fact_9637_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_9638_less__eq__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_eq_nat @ Xa @ X ) ) ).

% less_eq_natural.abs_eq
thf(fact_9639_less__eq__natural_Orep__eq,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X4: code_natural,Xa4: code_natural] : ( ord_less_eq_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_eq_natural.rep_eq
thf(fact_9640_less__natural_Orep__eq,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X4: code_natural,Xa4: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_natural.rep_eq
thf(fact_9641_less__natural__def,axiom,
    ( ord_le5570908160329646204atural
    = ( map_fu6256889081107267320ural_o @ code_nat_of_natural @ ( map_fu4892316939951275536at_o_o @ code_nat_of_natural @ id_o ) @ ord_less_nat ) ) ).

% less_natural_def
thf(fact_9642_less__eq__natural__def,axiom,
    ( ord_le1926595141338095240atural
    = ( map_fu6256889081107267320ural_o @ code_nat_of_natural @ ( map_fu4892316939951275536at_o_o @ code_nat_of_natural @ id_o ) @ ord_less_eq_nat ) ) ).

% less_eq_natural_def
thf(fact_9643_natural__decr,axiom,
    ! [N2: code_natural] :
      ( ( N2 != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N2 ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N2 ) ) ) ).

% natural_decr
thf(fact_9644_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9645_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9646_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9647_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T5: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S2 @ T5 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T5 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_9648_total__pair__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A3 @ fun_pair_less ) ).

% total_pair_less
thf(fact_9649_trans__pair__less,axiom,
    trans_4347625901269045472at_nat @ fun_pair_less ).

% trans_pair_less
thf(fact_9650_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_9651_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_9652_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_9653_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T5: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T5 ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_9654_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_9655_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T5: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S2 @ T5 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T5 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_9656_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_9657_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_9658_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_9659_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T5: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T5 ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_9660_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_9661_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_9662_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X6 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_9663_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_9664_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_9665_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_9666_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_9667_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_9668_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_9669_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_9670_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_9671_division__segment__int__def,axiom,
    ( euclid3395696857347342551nt_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% division_segment_int_def
thf(fact_9672_wmsI,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A3 ) @ ( set_ms8126754132646256062at_nat @ B3 ) ) @ fun_max_strict )
        | ( ( A3 = zero_z1048942125864253310at_nat )
          & ( B3 = zero_z1048942125864253310at_nat ) ) )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z4 @ A3 ) @ ( plus_p7104986032573967614at_nat @ Z4 @ B3 ) ) @ ms_weak ) ) ).

% wmsI
thf(fact_9673_smsI,axiom,
    ! [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A3 ) @ ( set_ms8126754132646256062at_nat @ B3 ) ) @ fun_max_strict )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z4 @ A3 ) @ ( plus_p7104986032573967614at_nat @ Z4 @ B3 ) ) @ ms_strict ) ) ).

% smsI
thf(fact_9674_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat @ ( produc5245064249948416855at_nat @ ms_strict @ ms_weak ) ).

% ms_reduction_pair
thf(fact_9675_ms__weak__def,axiom,
    ( ms_weak
    = ( sup_su3035147773818789531at_nat @ ms_strict @ id_mul2649389997224486051at_nat ) ) ).

% ms_weak_def
thf(fact_9676_ms__strictI,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z4 @ Z9 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A3 ) @ ( set_ms8126754132646256062at_nat @ B3 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z4 @ A3 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B3 ) ) @ ms_strict ) ) ) ).

% ms_strictI
thf(fact_9677_ms__weakI1,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z4 @ Z9 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A3 ) @ ( set_ms8126754132646256062at_nat @ B3 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z4 @ A3 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B3 ) ) @ ms_weak ) ) ) ).

% ms_weakI1
thf(fact_9678_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_9679_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X6: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( ( pw_leq @ X6 @ Y7 )
       => ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X6 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y7 ) ) ) ) ).

% pw_leq_step
thf(fact_9680_pw__leq_Osimps,axiom,
    ( pw_leq
    = ( ^ [A12: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
          ( ( ( A12 = zero_z1048942125864253310at_nat )
            & ( A22 = zero_z1048942125864253310at_nat ) )
          | ? [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,X11: multis2468970476368604999at_nat,Y8: multis2468970476368604999at_nat] :
              ( ( A12
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X4 @ zero_z1048942125864253310at_nat ) @ X11 ) )
              & ( A22
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y4 @ zero_z1048942125864253310at_nat ) @ Y8 ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ fun_pair_leq )
              & ( pw_leq @ X11 @ Y8 ) ) ) ) ) ).

% pw_leq.simps
thf(fact_9681_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A23: multis2468970476368604999at_nat] :
      ( ( pw_leq @ A1 @ A23 )
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A23 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X14: multis2468970476368604999at_nat] :
              ( ( A1
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X3 @ zero_z1048942125864253310at_nat ) @ X14 ) )
             => ! [Y9: multis2468970476368604999at_nat] :
                  ( ( A23
                    = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y3 @ zero_z1048942125864253310at_nat ) @ Y9 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
                   => ~ ( pw_leq @ X14 @ Y9 ) ) ) ) ) ) ).

% pw_leq.cases
thf(fact_9682_ms__weakI2,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z4 @ Z9 )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z4 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z9 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).

% ms_weakI2
thf(fact_9683_pw__leq__split,axiom,
    ! [X6: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
      ( ( pw_leq @ X6 @ Y7 )
     => ? [A8: multis2468970476368604999at_nat,B10: multis2468970476368604999at_nat,Z10: multis2468970476368604999at_nat] :
          ( ( X6
            = ( plus_p7104986032573967614at_nat @ A8 @ Z10 ) )
          & ( Y7
            = ( plus_p7104986032573967614at_nat @ B10 @ Z10 ) )
          & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A8 ) @ ( set_ms8126754132646256062at_nat @ B10 ) ) @ fun_max_strict )
            | ( ( B10 = zero_z1048942125864253310at_nat )
              & ( A8 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
thf(fact_9684_natLeq__on__well__order__on,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y4 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_9685_natLeq__on__Well__order,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ X4 @ N2 )
                & ( ord_less_nat @ Y4 @ N2 )
                & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y4 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_9686_cone__def,axiom,
    ( bNF_Cardinal_cone
    = ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).

% cone_def
thf(fact_9687_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_9688_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_9689_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_9690_natLeq__underS__less,axiom,
    ! [N2: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) ).

% natLeq_underS_less
thf(fact_9691_Restr__natLeq,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) )
          @ ^ [Uu2: nat] :
              ( collect_nat
              @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N2 ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N2 )
              & ( ord_less_nat @ Y4 @ N2 )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% Restr_natLeq

% Helper facts (51)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__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__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $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_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( if_mul8430962117462786573at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( if_mul8430962117462786573at_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__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] :
      ( ( if_opt7704869406773131885at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] :
      ( ( if_opt7704869406773131885at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( if_Pro4507677147265585453at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( if_Pro4507677147265585453at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( if_opt6883606601682554499it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( if_opt6883606601682554499it_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( if_opt1729522071442692626it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( if_opt1729522071442692626it_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    rep_assn @ ( times_times_assn @ ( q @ r2 ) @ r ) @ ( produc7507926704131184380et_nat @ h @ ( hoare_new_addrs @ h2 @ as @ h ) ) ).

%------------------------------------------------------------------------------